Existence and Exponential Behavior for the Stochastic 2D Cahn–Hilliard–Oldroyd Model of Order One

Abstract

In this article, we establish the unique global solvability of a 2D stochastic Cahn–Hilliard–Oldroyd model of order one, for the motion of an incompressible isothermal mixture of two (partially) immiscible non-Newtonian fluids having the same density and perturbed by a multiplicative noise of Gaussian and Lévy type. The model consists of the stochastic Oldroyd model of order one, coupled with a stochastic Cahn–Hilliard model. We prove the existence and uniqueness of a strong solution (in the stochastic sense). The proofs are based on the Galerkin approximation technique. Moreover, we also prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solutions.

1 Introduction

The Oldroyd model of order one arises in the dynamics of non-Newtonian fluids and is well-known as the generalization of the initial-boundary value problem for the Navier–Stokes equations. It is used to model the motion of viscoelastic fluids (see [23, 31]). The analysis of this model, both in the deterministic and the stochastic cases has been investigated by several authors (see for instance [23, 25,26,27, 31, 32]). The well-posedness and the exponential stability of the model in two-dimensional bounded and unbounded (Poincaré domains) domains, both in deterministic and stochastic settings is studied in [27]. The proof of the existence and the uniqueness of weak solution in the deterministic case is obtained via a local monotonicity property of the linear and nonlinear operators and a localized version of the Minty-Browder technique. The global solvability results for the stochastic counterpart are obtained by a stochastic generalization of the Minty-Browder technique. In order to describe the behavior of complex fluids in Fluid Mechanics, diffuse-interface methods are widely used by many researchers (see, e.g., [3, 15] and references therein). A typical example is a mixture of two incompressible fluids. The evolution of such a system is described by a sufficiently simple model so-called H model (see [19, 20, 30] and references therein). This consists in a suitable coupling of the Navier–Stokes equations for the (average) fluid velocity u through a capillarity force proportional to $\mu \nabla \phi$, where $\mu$ is the chemical potential, with a local or nonlocal Cahn–Hilliard type equation for the order parameter $\phi$ through a transport term $u\cdot \nabla \phi$. The mathematical and the numerical analysis of the deterministic and the stochastic local and nonlocal Cahn–Hilliard–Navier–Stokes (CH-NS) model has been considered by several authors such as [9,10,11,12,13,14, 17, 18, 21, 22, 24, 36,37,38] and references therein. In [13], the authors considered the stochastic 3D globally modified local CH–NS equations with multiplicative Gaussian noise. They proved the existence and uniqueness of strong solution (in the sense of partial differential equations and stochastic analysis) and derived the existence of a weak martingale solution for the stochastic 3D local CH-NS equations. The third author of the paper has proved the existence and uniqueness of the probabilistic strong solution for the stochastic 2D local CH-NS model with multiplicative Gaussian type of noise in [36] and studies the asymptotic stability of the unique probabilistic strong solution of 2D local CH-NS model in [37]. He also proved the existence of weak solution of 3D local CH-NS model with multiplicative non-Gaussian Lévy noise in [38]. Similar results have been obtained for the 2D and 3D nonlocal CH-NS model in [9, 11, 12]. Generally, the CH-NS model is used to modelise the flow of a Newtonian binary mixture. However, for viscoelastic binary fluids, the behavior of viscoelastic fluids cannot be predicted by the means of usual Newton’s constitutive law since they possess a memory of past deformations which is not the case for Newtonian fluids. Consequently, we have to introduce a more general phenomenological model such as Oldroyd model in addition to the Navier–Stokes equation as a constitutive equation to modelise the viscoelasticity. Taking into account this fact, we derive a model where we will call Cahn–Hilliard–Oldroyd model.

In this article, we study a stochastic generalization model of the CH-NS model. More precisely, we consider in a complete probability space $(\Omega ,{\mathcal {F}},{\mathbb {P}})$ equipped with an increasing family of sub-sigma fields $\{{\mathcal {F}}_t\}_{0\le t\le T}$ of ${\mathcal {F}}$ such that ${\mathcal {F}}_0$ contains all elements $F \in {\mathcal {F}}$ with ${\mathbb {P}}(F) = 0$ and ${\mathcal {F}}_t= \cap _{s>t}{\mathcal {F}}_s$ for $0 \le t \le T$, the following stochastic Cahn–Hilliard–Oldroyd model of order one, for the motion of an incompressible isothermal mixture of two immiscible non-Newtonian fluids

$$\begin{aligned} \left\{ \begin{array}{llc} du(t) +\left[ -\nu _1 \Delta u +(\beta *\Delta u)(t)+ (u.\nabla )u + \nabla p -{\mathcal {K}}\mu \nabla \phi \right] dt \\ \qquad \quad =\sigma _1(t,u,\phi )dW^1_t +\int _Z\gamma (t,u(t^-),\phi (t),z){\tilde{\pi }}(dt,dz), \\ d\phi (t)=\left[ \nu _2\Delta \mu -u.\nabla \phi \right] dt+\sigma _2(t,u,\phi )dW^2_t ,\\ \mu =-\varepsilon \Delta \phi +\alpha f(\phi ) , \\ {\text {div }}(u) = 0 , \end{array} \right. \end{aligned}$$

(1.1)

in $(0,T)\times {\mathcal {M}}$ with the conditions

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _\eta \phi =\partial _\eta \Delta \phi =0,\ \text { on } (0,T)\times \partial {\mathcal {M}}, \\ u=0 \ \text { on } (0,T)\times \partial {\mathcal {M}},\\ (u,\phi )(0)=(u_0,\phi _0) \ \text { in } {\mathcal {M}}, \end{array}\right. } \end{aligned}$$

(1.2)

where $T>0$, ${\mathcal {M}}$ is a bounded open domain in ${\mathbb {R}}^2$ with a smooth boundary $\partial {\mathcal {M}}$ and

$$\begin{aligned} (\beta *\Delta u)(t)=\int _0^t\beta (t-s)\Delta u(s)ds. \end{aligned}$$

In (1.1), $W^i_t$, $i=1,2$, are two cylindrical Wiener processes in a separable Hilbert space U defined on the probability space $(\Omega ,{\mathcal {F}},{\mathbb {P}})$. Also Z is a measurable subspace of some Hilbert space and ${\tilde{\pi }}(dt, dz):= \pi (dt, dz) - \lambda (dz)dt$ is a compensated Poisson random measure, where $\lambda (dz)$ is a $\sigma$-finite Lévy measure on the Hilbert space with an associated Poisson random measure $\pi (dt, dz)$ such that ${\mathbb {E}}[\pi (dt, dz)] = \lambda (dz)dt$. The processes $W^i_t$, $i=1,2$, and ${\tilde{\pi }}$ are mutually independent. Let $\nu > 0$ be the coefficient of kinematic viscosity. In (1.1), for $\varsigma ,\kappa > 0,$ we take

$$\begin{aligned} \nu _1=\frac{\kappa }{\varsigma }, \ \text { the kernel } \ \beta (t)=\gamma e^{-\delta t}, \ \text { where } \ \gamma =\frac{1}{\varsigma }(\nu -\frac{\kappa }{\varsigma })>0\ \text { and } \ \delta =\frac{1}{\varsigma }>0. \end{aligned}$$

(1.3)

If we take $\nu _1=\frac{\kappa }{\varsigma }$ (or $\gamma =0$) in (1.3), then we obtain a model for the phase separation of an incompressible and isothermal Newtonian binary fluid flow or the well-known Cahn–Hilliard–Navier–Stokes model [13, 37, 38]. Note that $\varsigma$ is the relaxation time, and is characterized by the fact that after instantaneous cessation of motion, the stresses in the fluid do not vanish instantaneously, but die out like $e^{-\varsigma ^{-t}}$. Moreover, the velocities of the flow, after instantaneous removal of the stresses, die out like $e^{-\kappa ^{-t}}$, where $\kappa$ is the retardation time. For the physical background and the mathematical modeling of viscoelastic fluid flows involving memory effects, we refer the interested readers to [2, 23, 34], where the topic is studied extensively. The quantity $\mu$ is the chemical potential of the binary mixture which is given by the variational derivative of the following free energy functional

$$\begin{aligned} {\mathcal {E}}_0(\phi )=\int _D\left( \frac{\varepsilon }{2}|\nabla \phi |^2+\alpha F(\phi )\right) dx, \end{aligned}$$

(1.4)

where, $F(r)=\int _0^rf(\zeta )d\zeta$ is the suitable double-well potential. The quantities $\nu _2$ and ${\mathcal {K}}$ are positive constants that correspond to the mobility constant and capillarity (stress) coefficient, respectively. $\varepsilon$ and $\alpha$ are two positive parameters describing the interactions between the two phases. In particular, $\varepsilon$ is related to the thickness of the interface separating the two fluids. A typical example of potential F is of logarithmic type. However, this potential is very often replaced by a polynomial approximation of the type $F(r)=\gamma _1r^4-\gamma _2r^2,$ with $\gamma _1$ and $\gamma _2$ are positive constants. Note that in (1.1), the $\phi$ equation modelises the evolution of the concentration of fluids which can be influenced by the thermal fluctuations which is a random phenomena. In order to take into account this thermal fluctuations, we have introduce a noise in addition to the $\phi$ equation as a constitutive equation. However, a Levy type noise in the $\phi$ equation is also possible but, the presence of such noise will involve probably more assumptions and will increase significantly the size of the paper.

To the best of our knowledge, there is no mathematical analysis of the model (1.1) even in its deterministic setting. This article is a contribution in that direction. Moreover, in the literature, there is no work on stochastic two-phase flows models with both Gaussian and non-Gaussian type of noise. But a such type of noise have been considered for instance in [27, 29] for the Oldroyd model of order one and the Navier–Stokes equation with hereditary viscosity. In general the presence of a noise on the concentration equation in the two-phase flow model makes the analysis of the model more involved (see [13, 16, 37]). In [13], an existence result has been obtained under the additional strong condition on the potential f. In order to use a weakened condition on the potential function f for the existence result, we have shown that the energy functional ${\mathcal {E}}_0$ is twice Fréchet differentiable which makes possible the application of Itô’s formula to the process ${\mathcal {E}}_0(\phi )$. Note that in (1.1) it is possible to add a Lévy type of noise on the relative concentration equation, but the analysis will be tedious. The purpose of the present manuscript is to prove some results related to problem (1.1). Our main results are the following: First, we prove the existence and uniqueness of strong solution (in the stochastic sense) for system (1.1). The method for the proof is based in the Galerkin approximation. Secondly, getting the existence of a unique strong solution in hand, we investigate the stability of this solution. More precisely, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solutions.

The rest of the paper is organized as follows. In the next section, we describe the mathematical setting required to establish the unique solvability of the system (1.1). The hypothesis satisfied by the potential, the noise coefficient and the external forcing are also discussed in the section. In Sect. 3, we introduce the Galerkin approximation of our problem and we derive a priori estimates for its solution. Then we prove the existence and the pathwise uniqueness of strong solution. In the last section, we analyze the stability of stationary solutions.

2 Functional Setting, Hypothesis and Abstract Formulation

In this section, we fix the hypothesis and describe the functional spaces needed to establish the existence and uniqueness of global strong solution of the system (1.1). We discuss the properties of linear and nonlinear operators, and also of the kernel $\beta (t) =\gamma e^{-\delta t}$.

2.1 Functional Setting

We now introduce the functional setup of Eqs. (1.1)–(1.2). If X is real Hilbert space with inner product $(.,.)_{X}$, then we denote the induced norm by $|.|_{X}$, while $X^*$ will indicate its dual. Let us consider the Hilbert spaces

$$\begin{aligned} H_{1}:=\overline{\left\{ u\in {\mathcal {C}}_{c}^{\infty }(({\mathcal {M}}))^2:\text {div}~u=0~\text {in}~ {\mathcal {M}}\right\} }^{{\mathbb {L}}^2},~ V_{1}:=\overline{\left\{ u\in {\mathcal {C}}_{c}^{\infty }(({\mathcal {M}}))^2:\text {div}~u=0~\text {in}~ {\mathcal {M}}\right\} }^{{\mathbb {H}}_{0}^1}, \end{aligned}$$

where ${\mathbb {L}}^2({\mathcal {M}}):=(L^2({\mathcal {M}}))^2$,  ${\mathbb {H}}_{0}^1({\mathcal {M}}):=(H_{0}^1({\mathcal {M}}))^2$. We endow $H_{1}$ with the $L^2-$inner product and norm

$$\begin{aligned} (u,v):=\int _{{\mathcal {M}}}u.v~dx,~|u|_{L^2}:=(u,u)^{1/2}. \end{aligned}$$

Moreover, the space $V_{1}$ is endowed with the scalar product and norm

$$\begin{aligned} ((u,v)):=\sum _{i=1}^{2}(\partial _{x_{i}}u,\partial _{x_{i}}v),~\Vert u\Vert =((u,u))^{1/2}. \end{aligned}$$

The norm in $V_{1}$ is equivalent to the ${\mathbb {H}}^1({\mathcal {M}})$-norm (due to Poincaré’s inequality). We refer the reader to [35] for more details on these spaces.

We now define the operator $A_{0}$ by

$$\begin{aligned} A_{0}u=-{\mathcal {P}}\Delta u,~\forall u\in D(A_{0})={\mathbb {H}}^2({\mathcal {M}})\cap V_{1}, \end{aligned}$$

where ${\mathcal {P}}$ is the Leray-Helmholtz projector in ${\mathbb {L}}^2({\mathcal {M}})$ onto $H_{1}$. Then, $A_{0}$ is a self-adjoint positive unbounded in $H_{1}$ which is associated with the scalar product defined above. Furthermore, $A_{0}^{-1}$ is a compact linear operator on $H_{1}$ and by the classical spectral theorem, there exists a sequence $\lambda _{j}$ with $0<\lambda _1<\lambda _2\le \cdots \le \lambda _n\le \lambda _{n+1}\le \cdots$ and a family $w_j\in D(A_0)$ which is an orthonormal basis in $H_1$ and such that $A_0w_j=\lambda _jw_j$.

For $u\in H_1$, we denote $u_j=(u,w_j)$. Given $\alpha >0$, take

$$\begin{aligned} D(A_0^\alpha )=\{u\in H_1: \sum _j\lambda _j^{2\alpha }|u_j|^2<\infty \}, \end{aligned}$$

(2.1)

and define $A_0^{\alpha }u=\sum _j\lambda _j^\alpha u_j w_j$, $u\in D(A_0^\alpha )$. We equip $D(A_0^\alpha )$ with the norm $|u|_{\alpha }^2:=|A_0^\alpha u|_{L^2}^2=\sum _j\lambda _j^{2\alpha }|u_j|^2$.

We introduce the linear nonnegative unbounded operator on $L^2({\mathcal {M}})$

$$\begin{aligned} A_{1}\varphi =-\Delta \varphi ,~\forall \varphi \in D(A_{1})=\left\{ \varphi \in H^{2}({\mathcal {M}}),~ \partial _{\eta }\varphi =0,~\text {on}~ \partial {\mathcal {M}}\right\} , \end{aligned}$$

(2.2)

and we endow $D(A_{1})$ with the norm $|A_{1}\cdot |+|\left\langle \cdot \right\rangle |$, which is equivalent to the $H^2$-norm. We also define the linear positive unbounded operator on the Hilbert space $L_{0}^2({\mathcal {M}})$ of the $L^2$-functions with null mean

$$\begin{aligned} B_{n}\varphi =-\Delta \varphi ,~\forall \varphi \in D(B_{n})=D(A_{1})\cap L_{0}^2({\mathcal {M}}). \end{aligned}$$

(2.3)

Note that $B_{n}^{-1}$ is a compact linear operator on $L_{0}^2({\mathcal {M}})$. More generally, we can define $B_{n}^s$, for any $s\in {\mathbb {R}}$, noting that $|B_{n}^{s/2}\cdot |_{L^2}$, $s>0$, is an equivalent norm to the canonical $H^s$-norm on $D(B_{n}^{s/2})\subset H^s({\mathcal {M}})\cap L_{0}^2({\mathcal {M}})$. Also note that $A_{1}=B_{n}$ on $D(B_{n})$. If $\varphi$ is such that $\varphi -\left\langle \varphi \right\rangle \in D(B_{n}^{s/2})$, we have that $|B_{n}^{s/2}(\varphi -\left\langle \varphi \right\rangle )|_{L^2}+|\varphi -\left\langle \varphi \right\rangle |_{L^2}$ is equivalent to the $H^s$-norm. Moreover, we set $H^{-s}({\mathcal {M}})=(H^s({\mathcal {M}}))'$, whenever $s<0$.

We note that

$$\begin{aligned} D(A_1)&=\left\{ \phi \in H^2: \frac{\partial \phi }{\partial \eta }=0~\text {on}~ \partial {\mathcal {M}}\right\} ,\nonumber \\ A_1\phi&=-\sum _{l=1}^{d}\frac{\partial ^2\phi }{\partial x_j^2},~\phi \in D(A_1). \end{aligned}$$

(2.4)

Classically, there exists a sequence $\beta _{j}$ with $0<\beta _1<\beta _2\le \cdots \le \beta _n\le \beta _{n+1}\le \cdots$ and a family $\psi _j\in D(A_1)$ which is an orthonormal basis in $L^2({\mathcal {M}})$ and such that $A_1\psi _j=\beta _j\psi _j$.

Now for $\alpha \ge 0$ we define

$$\begin{aligned} D(A_1^\alpha )=\{\phi \in H_2: \sum _j^\infty \beta _j^{2\alpha }|(\phi ,\psi _j)|^2<\infty \}, \end{aligned}$$

(2.5)

endowed with the Hilbertian norm $|\phi |_{\alpha }^2:=|A_1^\alpha \phi |_{L^2}^2=\sum _j\beta _j^{2\alpha }|(\phi ,\psi _j)|^2$.

We introduce the bilinear operators $B_{0}, B_{1}$ (and their associated trilinear forms $b_{0}, b_{1}$) as well as the coupling mapping $R_{0}$ which are defined, from $D(A_{0})\times D(A_{0})$ into $H_{1}$, $D(A_{0})\times D(A_{1})$ into $L^2({\mathcal {M}})$ and $L^2({\mathcal {M}})\times D(A_{1}^{3/2})$ into $H_{1}$, respectively. More precisely, we set

$$\begin{aligned} \begin{array}{lll} (B_{0}(u,v),w) &{} =\int _{{\mathcal {M}}}[(u \cdot \nabla )v] w dx \\ &{} =b_{0}(u,v,w),~\forall u,v, w\in D(A_{0}),\\ \\ (B_{1}(u,\varphi ),\psi )&{} =\int _{{\mathcal {M}}}[(u\cdot \nabla )\varphi ]\psi dx \\ &{} =b_{1}(u,\varphi ,\psi ),~ \forall u\in D(A_{0}), \varphi , \psi \in D(A_{1}),\\ \\ (R_{0}(\mu ,\varphi ),w) &{}=\int _{{\mathcal {M}}}[\mu \nabla \varphi ] w dx \\ &{} =b_{1}(w,\varphi ,\mu ),~\forall w\in D(A_{0}),~(\mu ,\varphi )\in L^2({\mathcal {M}})\times D(A_{1}^{3/2}). \end{array} \end{aligned}$$

(2.6)

Note that

$$\begin{aligned} R_{0}(\mu ,\varphi )={\mathcal {P}}\mu \nabla \varphi . \end{aligned}$$

Using an integration by part, we can check that

$$\begin{aligned} b_0(u,v,v)= & {} 0,~\forall u,v\in V_1,\nonumber \\ b_1(v,\phi ,\phi )= & {} 0,~\forall v\in V_1,~ \phi \in H^1({\mathcal {M}}), \end{aligned}$$

(2.7)

$$\begin{aligned} \left\langle R_0(A_1\phi ,\phi ),v\right\rangle= & {} \left\langle B_1(v,\phi ),A_1\phi \right\rangle =b_1(v,\phi ,A_1\phi ),~\forall (v,\phi )\in V_1\times D(A_1). \end{aligned}$$

(2.8)

We recall from [18] that, using the integration by parts, a suitable generalized Hölder inequality and a suitable Gagliardo-Nirenberg interpolation inequality, we derive that $b_0,$ and $b_1$ satisfy the following properties.

$$\begin{aligned} |b_0(u,v,w)|&\le c |u|_{L^2}^{1/2}\Vert u\Vert ^{1/2}\Vert v\Vert |w|_{L^2}^{1/2}\Vert w\Vert ^{1/2},~u, v,w\in V_1,\nonumber \\ |b_1(u,\phi ,\psi )|&\le c |u|_{L^2}^{1/2}\Vert u\Vert ^{1/2}\Vert \phi \Vert ^{1/2}|A_1\phi |_{L^2}^{1/2}|\psi |_{L^2}~u\in V_1,~ \phi \in D(A_1), ~\psi \in H_2,\nonumber \\ |b_1(u,\phi ,\psi )|&\le c |u|_{L^2}\Vert \phi \Vert ^{1/2}|A_1\phi |_{L^2}^{1/2}\Vert \psi \Vert ~u\in V_1,~ \phi \in D(A_1), ~\psi \in H_2. \end{aligned}$$

(2.9)

$$\begin{aligned} \Vert B_0(u,v)\Vert _{V_1^*}&\le c |u|_{L^2}^{1/2}\Vert u\Vert ^{1/2}|v|_{L^2}^{1/2}\Vert v\Vert ^{1/2}~u, v,w\in V_1. \end{aligned}$$

(2.10)

$$\begin{aligned} |R_0(A_1\phi ,\rho )|_{V_1^{*}}&\le c{\left\{ \begin{array}{ll} \Vert \rho \Vert ^{1/2}|A_1\rho |_{L^2}^{1/2}|A_1\phi |_{L^2},~\phi , \rho \in D(A_1)\\ \Vert \rho \Vert ^{1/2}|A_1\rho |_{L^2}^{1/2}\left\| \phi \right\| ^{1/2}|A^{3/2}_1\phi |^{1/2}_{L^2},~ \rho \in D(A_1),~\phi \in D(A_1^{3/2}). \end{array}\right. } \end{aligned}$$

(2.11)

$$\begin{aligned} \Vert B_1(u,\phi )\Vert _{V_2^*}&\le c |u|_{L^2}\Vert \phi \Vert ^{1/2}|A_1\phi |_{L^2}^{1/2},~u\in H_1,~ \phi \in D(A_1). \end{aligned}$$

(2.12)

owing to (1.2)$_1$ we derive that

$$\begin{aligned} \partial _\eta \mu =0\ \text { on } \ (0,T)\times \partial {\mathcal {M}}. \end{aligned}$$

(2.13)

From (2.13), we deduce the mass conservation in he deterministic case. In fact, for $\sigma _2(\cdot )=0$, from (2.13), we have the conservation of the following quantity

$$\begin{aligned} \langle \phi (t) \rangle =\left| {\mathcal {M}}\right| ^{-1}\int _{\mathcal {M}}\phi (t,x)dx, \end{aligned}$$

(2.14)

where $\left| {\mathcal {M}}\right|$ stands for the Lebesgue measure of the domain ${\mathcal {M}}$. More precisely, we have

$$\begin{aligned} \langle \phi (t) \rangle =\langle \phi (0) \rangle ,\quad \forall t\in [0,T]. \end{aligned}$$

(2.15)

Hereafter, we assume that $\sigma _2(\cdot )$ is chosen such that (2.14) is satisfied, which is the case if we assume that

$$\begin{aligned} \left\langle \int _0^t\sigma _2(s,v,\psi )dW^2_s\right\rangle =0,\quad \forall t\ge 0,\ v\in H_1,\ \psi \in H_2, \end{aligned}$$

(2.16)

where $H_2$ is defined by (2.19) below.

Due to the mass conservation, we have

$$\begin{aligned} \left\langle \phi (t)\right\rangle =\left\langle \phi (0)\right\rangle =:M_{0},~\forall t\ge 0. \end{aligned}$$

(2.17)

Thus, up to a shift of the order parameter field, we can always assume that the mean of $\phi$ is zero a the initial time and, therefore it will remain zero for all positive times. Hereafter, we assume that

$$\begin{aligned} \left\langle \phi (t)\right\rangle =\left\langle \phi (0)\right\rangle =0,~\forall t>0. \end{aligned}$$

(2.18)

We set

$$\begin{aligned} H_2=D(A_{1}^{1/2})\quad \text {and}\quad {\mathcal {H}}=H_{1}\times D(A_{1}^{1/2}). \end{aligned}$$

(2.19)

The norm in $H_2$ is denoted by $\Vert \cdot \Vert$, where $\Vert \psi \Vert ^2=|A_1^{1/2}\psi |_{L^2}^2$. The space ${\mathcal {H}}$ is a complete metric space with respect to the metric associated with the norm

$$\begin{aligned} |(v,\psi )|_{{\mathcal {H}}}^2={\mathcal {K}}^{-1}|v|^2+\epsilon \Vert \psi \Vert ^2. \end{aligned}$$

(2.20)

We set $V_2=D(A_1)$ and define the Hilbert spaces ${\mathcal {U}}$ and ${\mathbb {V}}$ respectively by

$$\begin{aligned} {\mathcal {U}}=V_{1}\times D(A^{3/2}_{1}),\qquad {\mathbb {V}}=V_{1}\times V_2=V_1\times D(A_1), \end{aligned}$$

(2.21)

endowed with the scalar product whose associated norm are respectively

$$\begin{aligned} \Vert (v,\psi )\Vert _{{\mathcal {U}}}^2=\Vert v\Vert ^2+|A^{3/2}_{1}\psi |_{L^2}^2,\qquad \Vert (v,\psi )\Vert _{{\mathbb {V}}}^2=\Vert v\Vert ^2+|A_{1}\psi |_{L^2}^2. \end{aligned}$$

(2.22)

We will denote by $\lambda _0$ a positive constant such that

$$\begin{aligned} \lambda _0|(v,\psi )|_{{\mathcal {H}}}^2\le \Vert (v,\psi )\Vert _{{\mathcal {U}}}^2\quad \forall (v,\psi )\in {\mathcal {U}}. \end{aligned}$$

(2.23)

For $u_1=(v_1,\phi _1),$ $u_2=(v_2,\phi _2),$ $u_3=(v_3,\phi _3)\in {\mathbb {V}},$ we define

$$\begin{aligned}&\langle B(u_1,u_2),u_3\rangle =b(u_1,u_2,u_3)=b_0(v_1,v_2,v_3)+b_1(u_1,\phi _2,\phi _3),\\&R(u_1,u_2)=(R_0(A_1\phi _1,\phi _2),0),\quad E(u_1)=(0,A_1f(\phi _1)). \end{aligned}$$

Lemma 2.1

The maps B,  R and E are locally Lipschitz continuous i.e. for every $r>0,$ there exists a constant $C_r$ such that

$$\begin{aligned}&\left\| B(v_1,v_1)-B(v_2,v_2)\right\| _{{\mathbb {V}}^{*}}\le C_r\left\| v_1-v_2\right\| _{\mathbb {V}},\nonumber \\&\left\| R(v_1,v_1)-R(v_2,v_2)\right\| _{{\mathbb {V}}^{*}}\le C_r\left\| v_1-v_2\right\| _{\mathbb {V}}, \nonumber \\&\left\| E(v_1)-E(v_2)\right\| _{{\mathbb {V}}^{*}}\le C_r\left\| v_1-v_2\right\| _{\mathbb {V}}, \end{aligned}$$

(2.24)

for all $v_1=(u_1,\phi _1),$ $u_2=(u_2,\phi _2)\in {\mathbb {V}}$ with $\left\| v_1\right\| _{\mathbb {V}}$ and $\left\| v_2\right\| _{\mathbb {V}}\le r$, where ${\mathbb {V}}^{*}$ is the dual space of ${\mathbb {V}}$.

Proof

Let $v_1=(u_1,\phi _1),$ $u_2=(u_2,\phi _2)\in {\mathbb {V}}$ and $(w,\psi )\in {\mathbb {V}}.$ We assume that $\left\| v_1\right\| _{\mathbb {V}}\le r$ and $\left\| v_2\right\| _{\mathbb {V}}\le r.$ To prove (2.24)$_1$, we note that

$$\begin{aligned} \left\| B(v_1,v_1)-B(v_2,v_2)\right\| _{{\mathbb {V}}^{*}}=\left\| B_0(u_1,u_1)-B_0(u_2,u_2)\right\| _{V_1^{*}}+\left\| B_1(u_1,\phi _1)-B(v_2,\phi _2)\right\| _{D(A_1^{-1})}. \end{aligned}$$

Also,

$$\begin{aligned}&\left| \langle B_0(u_1,u_1)-B_0(u_2,u_2),w \rangle \right| =\left| b_0(u_1,u_1,w)-b_0(u_1,u_2,w)+b_0(u_1,u_2,w)-b_0(u_2,u_2,w)\right| \\&\quad =\left| b_0(u_1-u_2,u_2,w)+b_0(u_1,u_1-u_2,w)\right| \le \left| b_0(u_2-u_1,u_2,w)\right| +\left| b_0(u_1,u_2-u_1,w)\right| \\&\quad \le c\left\| u_1-u_2\right\| \left\| u_2\right\| \left\| w\right\| +c\left\| u_1\right\| \left\| u_1-u_2\right\| \left\| w\right\| =2c\left\| u_2-u_1\right\| \left\| w\right\| , \end{aligned}$$

which implies that

$$\begin{aligned} \left\| B_0(u_1,u_1)-B_0(u_2,u_2)\right\| _{V_1^{*}}\le 2c \left\| u_2-u_1\right\| . \end{aligned}$$

(2.25)

Proceeding similarly as in (2.25) and using the fact that $D(A_1)$ is continuously embedded in $L^2({\mathcal {M}})$, we obtain

$$\begin{aligned}&\left| \langle B_1(u_1,\phi _1)-B_1(u_2,\phi _2),\psi \rangle \right| \le \left| b_1(u_1-u_2,\phi _1,\psi )\right| +\left| b_1(u_2,\phi _1-\phi _2,\psi )\right| \\&\quad \le c\left\| u_1-u_2\right\| \left| A_1\phi _1\right| _{L^2}\left| A_1\psi \right| _{L^2}+c\left\| u_2\right\| \left| A_1\psi \right| _{L^2}\left| A_1(\phi _1-\phi _2)\right| _{L^2}\\&\quad \le cr(\left\| u_1-u_2\right\| +\left| A_1(\phi _1-\phi _2)\right| _{L^2})\left| A_1\psi \right| _{L^2}. \end{aligned}$$

It then follows that

$$\begin{aligned} \left\| B_1(u_1,\phi _1)-B(v_2,\phi _2)\right\| _{D(A_1^{-1})}\le cr(\left\| u_1-u_2\right\| +\left| A_1(\phi _1-\phi _2)\right| _{L^2}). \end{aligned}$$

(2.26)

From (2.25) and (2.26) we derive that

$$\begin{aligned} \left\| B(v_1,v_1)-B(v_2,v_2)\right\| _{{\mathbb {V}}^{*}}&\le 2c \left\| u_2-u_1\right\| \left\| w\right\| + cr(\left\| u_1-u_2\right\| +\left| A_1(\phi _1-\phi _2)\right| _{L^2})\\&\le C_r\left\| v_1-v_2\right\| _{\mathbb {V}} \end{aligned}$$

which prove (2.24)$_1.$ Now, remark that

$$\begin{aligned} \left\| R(v_1,v_1)-R(v_2,v_2)\right\| _{{\mathbb {V}}^{*}}= \left\| R_0(A_1\phi _1,\phi _1)-R(A_1\phi _2,\phi _2)\right\| _{V_1^{*}}. \end{aligned}$$

However,

$$\begin{aligned}&\left| \langle R_0(A_1\phi _1,\phi _1)-R(A_1\phi _2,\phi _2),w \rangle \right| =\left| b_1(w,\phi _1,A_1(\phi _1-\phi _2))\right| +\left| b_1(w,\phi _1-\phi _2,A_1\phi _2)\right| \\&\quad \le c\left\| w\right\| \left| A_1\phi _1\right| _{L^2}\left| A_1(\phi _1-\phi _2)\right| _{L^2}+c\left\| w\right\| \left| A_1\phi _2\right| _{L^2}\left| A_1(\phi _1-\phi _2)\right| _{L^2}\\&\quad \le cr\left\| w\right\| \left| A_1(\phi _1-A_1\phi _2)\right| _{L^2}. \end{aligned}$$

It follows that

$$\begin{aligned} \left\| R_0(A_1\phi _1,\phi _1)-R(A_1\phi _2,\phi _2)\right\| _{V_1^{*}}\le C_r\left| A_1(\phi _1-A_1\phi _2)\right| _{L^2}\le C_r\left\| v_1-v_2\right\| _{\mathbb {V}}, \end{aligned}$$

which prove (2.24)$_2.$ Also, we remark that

$$\begin{aligned} \left\| E(v_1)-E(v_2)\right\| _{{\mathbb {V}}^{*}}=\left\| A_1f(\phi _1)-A_1f(\phi _2)\right\| _{D(A_1^{-1})}. \end{aligned}$$

Let us recall from [18] that, there exists a monotone non-decreasing function $Q_1(x_1,x_2)$ of $x_1$ and $x_2$ such that

$$\begin{aligned} \begin{array}{ll} \left| \langle A_1f(\phi _1)-A_1f(\phi _2), \psi \rangle \right| &{} =\left| \langle f(\phi _1)-f(\phi _2), A_1 \psi \rangle \right| \\ &{} \le Q_1(\left\| \phi _1\right\| ,\left\| \phi _2\right\| )\left| A_1(\phi _1-\phi _2)\right| _{L^2}\left| A_1\psi \right| _{L^2}. \end{array} \end{aligned}$$

We recall that, since $\left\| \phi _1\right\| \le c\left| A_1\phi _1\right| _{L^2}\le cr,$ $\left\| \phi _2\right\| \le c\left| A_1\phi _2\right| _{L^2}\le cr$ and $Q_1$ is a monotone non-decreasing function, we have $Q_1(\left\| \phi _1\right\| ,\left\| \phi _2\right\| )\le Q_1(cr,cr).$ So, we obtain

$$\begin{aligned} \left| \langle A_1f(\phi _1)-A_1f(\phi _2), \psi \rangle \right| \le Q_1(cr,cr)\left| A_1(\phi _1-\phi _2)\right| _{L^2}\left| A_1\psi \right| _{L^2}. \end{aligned}$$

Hence

$$\begin{aligned} \begin{array}{ll} \left\| E(v_1)-E(v_2)\right\| _{{\mathbb {V}}^{*}} &{} =\left\| A_1f(\phi _1)-A_1f(\phi _2)\right\| _{D(A_1^{-1})} \\ &{} \le Q_1(cr,cr)\left| A_1(\phi _1-\phi _2)\right| _{L^2}\le C_r\left\| v_1-v_2\right\| _{\mathbb {V}}, \end{array} \end{aligned}$$

which proves (2.24)$_3$ and ends the proof of Lemma 2.1. $\square$

Now, we discuss some properties of a general kernel $\beta (.)$ and then in particular $\beta (t)=\gamma e^{-\delta t}$. we define

$$\begin{aligned} (Lu)(t)=(\beta *u)(t)=\int _0^t\beta (t-s)u(s)ds. \end{aligned}$$

A function $\beta (\cdot )$ is called positive kernel if the operator L is positive on $L^2(0,T;H_1)$ for all $T>0$. That is,

$$\begin{aligned} \int _0^T(Lu(t),u(t))=\int _0^T\int _0^t\beta (t-s)(u(s),u(s))dsdt\ge 0, \end{aligned}$$

for all $u\in L^2(0,T;H_1)$ and every $T>0$.

Let ${\hat{\beta }}(\theta )$ be the Laplace transform of $\beta (t)$, i.e.

$$\begin{aligned} {\hat{\beta }}(\theta )=\int _0^\infty e^{-\theta r}\beta (r)dr,\qquad \theta \in {\mathbb {C}}. \end{aligned}$$

Then we have from [5, Lemma 4.1] the following result.

Lemma 2.2

Let $\beta \in L^\infty (0,\infty )$ be such that $Re {\hat{\beta }}(\theta )>0$ if $Re(\theta )>0$. Then, $\beta (t)$ defines a positive kernel.

We also recall from [29, Lemma A$_1$] the following result.

Lemma 2.3

Let $\beta (t)=\gamma e^{-\delta t}$, $\delta >0$, $t\in [0,T]$. Then for any right continuous function with left limits, $f:[0,T]\rightarrow [0,\infty )$, we have

$$\begin{aligned} \int _0^Tf(t)((\beta *u)(s), u(s))ds\ge 0, \end{aligned}$$

(2.27)

for all $u\in L^2(0,T; H_1)$.

Remark 2.1

• (1) As proved in [28, Lemma 2.6], with a change of variable and change of integrals, it can be easily seen that, if $\beta \in L^1(0,T)$, $f,g\in L^2(0,T)$, for some $T>0$, then we have

  $$\begin{aligned} \left( \int _0^Tg^2(t)dt\left( \int _0^t\beta (t-s)f(s)ds\right) ^2\right) ^{1/2}\le \left( \int _0^T\left| \beta (t)\right| dt\right) \left( \int _0^Tg^2(t)f^2(t)dt\right) ^{1/2}. \end{aligned}$$

  (2.28)

• (2) If we take $g(t)=1$, and $f(t)=\left| u(t)\right| _{L^2}$ in (2.28) with $u\in L^2(0,T;H_1)$, we obtain

  $$\begin{aligned} \left( \int _0^T\left( \int _0^t\beta (t-s)\left| u(s)\right| ^2_{L^2}ds\right) ^2\right) ^{1/2}\le \left( \int _0^T\left| \beta (t)\right| dt\right) \left( \int _0^T\left| u(t)\right| ^2_{L^2}dt\right) ^{1/2}. \end{aligned}$$

  (2.29)

• (3) For $\beta (t)=\gamma e^{-\delta t}$, we know that

  $$\begin{aligned} \int _0^\infty \beta (t)dt=\frac{\gamma }{\delta }\ \text { and } \ {\hat{\beta }}(\theta )=\frac{\gamma }{\theta +\delta }>0,\ \text { for } \ Re\theta >0, \end{aligned}$$

  and by Lemma 2.2, $\beta (t)$ is a positive kernel. Hence, we have

  $$\begin{aligned} \int _0^T\langle (\beta *A_0u)(t), u(t) \rangle dt=\int _0^T( (\beta *\nabla u)(t),\nabla u(t) )dt\ge 0. \end{aligned}$$

  (2.30)

  Using the fact that $\left\| A_0u\right\| _{V_1^*} \le \left\| u\right\|$, we get

  $$\begin{aligned} \left\| (\beta *A_0u)(t)\right\| _{V_1^*}\le \int _0^t\beta (t-s)\left\| A_0u(s)\right\| _{V_1^*}ds\le \int _0^t\beta (t-s)\left\| u(s)\right\| ds, \end{aligned}$$

  (2.31)

  and hence by (2.29), we have

  $$\begin{aligned} \int _0^T\left\| (\beta *A_0u)(t)\right\| ^2_{V_1^*}dt&\le \int _0^T\left( \int _0^t\beta (t-s)\left\| A_0u(s)\right\| _{V_1^*}ds\right) ^2dt\nonumber \\&\le \left( \int _0^T\beta (t)dt\right) ^2\int _0^T\left\| u(t)\right\| ^2dt\le \frac{\gamma ^2}{\delta ^2}\int _0^T\left\| u(t)\right\| ^2dt, \end{aligned}$$

  (2.32)

  for $u\in L^2(0,T;V_1)$.

Remark 2.2

Using the Cauchy-Schwarz and Hölder inequalities and (2.32), we derive that

$$\begin{aligned} \int _0^T\langle (\beta *A_0u)(t), u(t) \rangle dt&\le \left( \int _0^t\left\| (\beta *A_0u)(s)\right\| ^2_{V_1^*}ds\right) ^{1/2}\left( \int _0^t\left\| u(s)\right\| ^2ds\right) ^{1/2}\nonumber \\&\le \left( \int _0^t\beta (s)ds\right) \int _0^t\left\| u(s)\right\| ^2ds\le \frac{\gamma }{\delta }\int _0^t\left\| u(s)\right\| ^2ds. \end{aligned}$$

(2.33)

2.2 Hypothesis

We assume that $W^i$, $i = 1,2$ are formally given by the expansion

$$\begin{aligned} W^i(t)=\sum _{j=1}^\infty \beta _j(t)e_j,\quad j=1,2, \end{aligned}$$

(2.34)

where $\beta _j(t)$, $j \in {\mathbb {N}}$ are independent one dimensional Brownian motions on $(\Omega , {\mathcal {F}}, {\mathbb {P}})$, and $\{ \beta _j\}_{j=1}^\infty$ is an orthonormal basis on U. We also define the auxiliary space $U_0$ containing U, that is defined by

$$\begin{aligned} U_0=\left\{ \sum _{j=1}^\infty \alpha _je_j:\quad \sum _{j=1}^\infty \frac{\alpha _j^2}{j^2}<\infty \right\} , \end{aligned}$$

endowed with the scalar product

$$\begin{aligned} (u,v)_{U_0}=\sum _{j=1}^\infty \frac{\alpha _j\beta _j}{j^2},\quad \text {for } u=\sum _{j=1}^\infty \alpha _je_j, \ v=\sum _{j=1}^\infty \beta _je_j. \end{aligned}$$

The stochastic forcing takes the following form

$$\begin{aligned} \sigma _i(t,u,\phi )dW^i(t)=\sum _{j=1}^\infty \sigma _j^i(t,u,\phi )d\beta _j(t),\quad i=1,2, \end{aligned}$$

(2.35)

with suitable restrictions on the growth of the diffusion coefficients $\sigma _j^i$ specified below.

Let us denote by ${\mathbf {D}}([0, T ]; H_1)$, the set of all $H_1$-valued functions defined on [0, T], which are right continuous and have left limits (Càdlàg functions) for every $t \in [0, T ]$. Also, let

$$\begin{aligned} {\mathcal {M}}_T^{2p}(H_1)=L^{2p}(\Omega \times (0,T]\times Z,{\mathcal {B}}((0,T]\times {\mathcal {F}}\times Z ), dt\otimes {\mathbb {P}}\otimes \lambda ; H_1), \end{aligned}$$

(2.36)

be the space of all ${\mathcal {B}}((0,T]\times {\mathcal {F}}\times Z )$ measurable functions $\gamma :[0,T]\times \Omega \times Z\rightarrow H_1$ such that

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T\int _Z\left| \gamma (t,\cdot ,z)\right| ^{2p}_{L^2}\lambda (dz)dt \right] <+\infty . \end{aligned}$$

For any Hilbert space H, we will denote by ${\mathcal {L}}^2(U;H)$ the separable Hilbert space of Hilbert-Schmidt operators from U into H.

To simplify the notations, we set (without loss of generality) $\nu _1 = \nu _2 = \varepsilon = \alpha = {\mathcal {K}}= 1$. Let us assume that the potential function f and the noise coefficients $\sigma _i(\cdot , \cdot )$, $i=1,2$ and $\gamma (\cdot , \cdot , \cdot )$ satisfy the following hypothesis.

(H1):

$f\in {\mathcal {C}}^{2}({\mathbb {R}})$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \liminf _{|r|\rightarrow +\infty } f'(r)>0, \\ |f^{(i)}(r)|\le c_{f}(1+|r|^{2-i}),~\forall r\in {\mathbb {R}},~i=0,1,2, \end{array}\right. } \end{aligned}$$

(2.37)

where $c_{f}$ is some positive constant.

(H2):

For all $t\in [0,T]$, $\langle \int _0^t\sigma _2(s,u,\phi )dW^2_s\rangle =0$ and there exist positive constants $K_0$ ; $K_1$ such that

$$\begin{aligned} \begin{aligned} \int _Z\left| \gamma (t,u,\phi ,z)\right| _{L^2}^2\lambda (dz)&\le K_0(1+\left| (u,\phi )\right| ^2_{{\mathcal {H}}}),\\ \int _Z\left| \gamma (t,u,\phi ,z)\right| _{L^2}^4\lambda (dz)&\le K_1(1+\left| (u,\phi )\right| ^4_{{\mathcal {H}}}), \\ \left| \sigma _2(t,u,\phi )\right| ^2_{{\mathcal {L}}^2(U;H_2)}&=\sum _{j=1}^\infty \left\| \sigma _j^2(t,u,\phi )\right\| ^2\le K_0, \end{aligned} \end{aligned}$$

(2.38)

uniformly in $t \in [0,T ]$ for all $(u,\phi )\in {\mathcal {H}}$.

(H4):

For all $t\in [0,T]$, there exists a positive constant L such that

$$\begin{aligned}&\sum _{i=1}^{2}\left| \sigma _i(t,u_1,\phi _1)-\sigma _i(t,u_2,\phi _2)\right| ^2_{{\mathcal {L}}^2(U;H_i)}+\int _Z\left| \gamma (t,u_1,\phi _1,z)-\gamma (t,u_2,\phi _2,z)\right| _{L^2}^2\lambda (dz)\nonumber \\&\qquad \le L\left| (u_1,\phi _1)-(u_2,\phi _2)\right| _{{\mathcal {H}}}^2, \end{aligned}$$

(2.39)

for all $(u_i,\phi _i)\in {\mathcal {H}}$, $i=1,2$.

Remark 2.3

Condition (2.38)$_3$ on the noise is widely employed in literature (see [16]). This condition implies that $\sigma _2(.,0,0)\in L^p(\Omega ,{\mathcal {F}},{\mathbb {P}};L^2(0,T;{\mathcal {L}}^2(U;H_2)))$, for all $p\ge 2$. Indeed,

$$\begin{aligned} {\mathbb {E}}\left( \int _0^T\left\| \sigma _2(.,0,0)\right\| _{{\mathcal {L}}^2(U;H_2)}^2ds\right) ^{p/2}\le K_0^{p/2}T^{p/2}<\infty . \end{aligned}$$

For any $(v,\psi )\in {\mathcal {H}},$ we set

$$\begin{aligned} {\mathcal {E}}(v,\psi )=\left| v\right| ^2_{L^2}+2{\mathcal {E}}_0(\psi )+c_1=\left| (v,\psi )\right| ^2_{\mathcal {H}}+2\alpha \langle F(\psi ),1\rangle +c_1, \end{aligned}$$

(2.40)

where $c_1>0$ is a constant large enough and independent on $(v,\psi )$ such that ${\mathcal {E}}(v,\psi )$ is non-negative (note that $F_0$ is bounded from below).

2.3 Abstract Formulation

Using the notations above, we rewrite problem (1.1)–(1.2) as:

$$\begin{aligned} {\left\{ \begin{array}{ll} du(t)+ [\nu _1A_{0}u+(\beta *A_0u)(t)+B_{0}(u,u)-{\mathcal {K}}R_{0}(A_{1}\phi ,\phi )]dt\\ =\sigma _1(t,u,\phi )dW^1_t +\int _Z\gamma (t,u(t^-),\phi (t),z){\tilde{\pi }}(dt,dz),~ \text {in}~ V^*_{1}, \\ d\phi (t)+[\nu _2 A_{1}\mu +B_{1}(u,\phi )]dt=\sigma _2(t,u,\phi )dW^2_t,~ \text {in}~H^{-1}(D),\\ \mu =-\varepsilon A_1 \phi +\alpha f(\phi ) ,~ \text {in}~H^{-1}(D),\\ (u,\phi )(0)=(u_{0},\phi _{0}), \end{array}\right. } \end{aligned}$$

(2.41)

which is equivalent to for all $(v,\psi )\in V_1\times H^1(D)$,

$$\begin{aligned} \begin{aligned}&(u(t),v)+ \int _0^t\langle \nu _1A_{0}u+(\beta *A_0u)(t)+B_{0}(u,u)-{\mathcal {K}}R_{0}( A_{1}\phi ,\phi ), v\rangle dt =(u_0,v) \\&\qquad +\int _0^t(\sigma _1(s,u,\phi )dW^1_s,v) +\int _0^t\int _Z(\gamma (s,u(s^-),\phi (s),z),v){\tilde{\pi }}(ds,dz), \\&\quad \qquad (\phi (t),\phi )+\int _0^t\langle \nu _2\nabla \mu ,\nabla \psi \rangle ds= (\phi _0,\psi )+\int _0 ^t(\sigma _2(s,u,\phi )dW^2_s,\psi ),\\&\qquad \mu =-\varepsilon A_1 \phi +\alpha f(\phi ) \end{aligned} \end{aligned}$$

(2.42)

${\mathbb {P}}$-a.s. and for all $t\in [0,T]$, for some fixed point $(u_0,\phi _0)$ in ${\mathcal {H}}$.

Remark 2.4

In the weak formulation (2.41), the term $\mu \nabla \phi$ is replaced by $A_{1}\phi \nabla \phi$. This is justified since $f(\phi )\nabla \phi$ is the gradient of $F(\phi )$ and can be incorporated into the pressure gradient, see [18] for details.

Let us now give the definition of a unique global strong solution in the probabilistic sense to the system (2.41).

Definition 2.1

(Global strong solution) Let the ${\mathcal {F}}_0$-measurable initial data $(u_0,\phi _0)\in L^4(\Omega ,{\mathcal {F}},{\mathbb {P}};{\mathcal {H}})$ be given. An ${\mathcal {H}}$-valued ${\mathcal {F}}_t$-adapted càdlàg process $(u,\phi )(\cdot )$ is called a strong solution to (2.41) if $(u,\phi )\in L^p(\Omega ,{\mathcal {F}},{\mathbb {P}}; L^\infty (0,T;{\mathcal {H}}))\cap L^p(\Omega ,{\mathcal {F}},{\mathbb {P}}; L^2(0,T;{\mathcal {U}})),$ for all $p\ge 2$ and satisfies (2.42).

Definition 2.2

A strong solution $(u,\phi )(\cdot )$ to (2.41) is called a unique strong solution if $({\tilde{u}},{\tilde{\phi }})(\cdot )$ is an another strong solution, then

$$\begin{aligned} {\mathbb {P}}\left\{ \omega \in \Omega ;\ (u,\phi )(t)=({\tilde{u}},{\tilde{\phi }})(t),\ \text { for all } \ t\in [0,T]\right\} =1. \end{aligned}$$

3 Existence and Uniqueness

In this section, we establish the global solvability of the system (2.41). To simplify the notations, throughout this section, we will set (without loss of generality) $\nu _1=\nu _2=\alpha =\varepsilon ={\mathcal {K}}=1$. We first prove the following energy type equality.

Proposition 3.1

If $(u, \phi )$ is a variational solution to (2.41), then $(u,\phi )$ satisfies

$$\begin{aligned} {\mathcal {E}}(u,\phi )(t)&+\int _0^t ((\beta *\nabla u)(s), \nabla u(s)) ds+2\int _0^t\left( \left\| u(s)\right\| ^2+\left| \nabla \mu (s)\right| _{L^2}^2\right) ds={\mathcal {E}}(u_0,\phi _0)\nonumber \\&+\int _0^t\left| \sigma _1(s,u,\phi )\right| ^2_{{\mathcal {L}}^2(U;H_1)}ds +2\int _0^t(\sigma _1(s,u,\phi ),u)dW^1(s)+2\int _0^t(\sigma _2(s,u,\phi ),\mu )dW^2(s)\nonumber \\&\quad \int _0^t\sum _{j=1}^{+\infty }\int _{\mathcal {M}}\left[ \left| \nabla \sigma _j^2(s,u,\phi )(x)\right| ^2+\left| f'(\phi (s,x))\right| \left| \sigma _j^2(s,u,\phi )(x)\right| ^2 \right] dxds\nonumber \\&+2\int _0^t\int _{Z}(\gamma (s,u(s^-),\phi (s),z),u){\tilde{\pi }}(ds,dz)+\int _0^t\int _{Z}\Upsilon (s,z)\pi (ds,dz), \end{aligned}$$

(3.1)

where

$$\begin{aligned} \Upsilon (s,z)=\left| u(s^-)+\gamma (s,u(s^-),\phi (s),z)\right| ^2_{L^2}-\left| u(s^-)\right| ^2_{L^2}-2(\gamma (s,u(s^-),\phi (s),z),u(s)). \end{aligned}$$

Proof

We apply infinite dimensional Itô’s formula (see [33]) to the process $\left| u\right| _{L^2}^2$ to find

$$\begin{aligned} \left| u(t)\right| ^2_{L^2}&=\left| u(0)\right| ^2_{L^2}-2\int _0^t\langle A_{0}u+(\beta *A_0u)(s)-R_{0}(A_{1}\phi ,\phi ), u\rangle ds\nonumber \\&\quad +\int _0^t\left| \sigma _1(s,u,\phi )\right| ^2_{{\mathcal {L}}^2(U;H_1)}ds+2\int _0^t(\sigma _1(s,u,\phi ),u)dW^1(s)\nonumber \\&\quad +2\int _0^t\int _{Z}(\gamma (s,u(s^-),\phi (s),z),u){\tilde{\pi }}(ds,dz)+\int _0^t\int _{Z}\Upsilon (s,z)\pi (ds,dz). \end{aligned}$$

(3.2)

We want now to write Itô’s formula for the free energy functional ${\mathcal {E}}_0(\phi )$, $\phi \in D(A_1)$. To this end, we should first prove that ${\mathcal {E}}_0: D(A_1)\rightarrow [0,\infty )$ is twice Fréchet differentiable. Let $\phi ,\psi \in D(A_1)$, using the Taylor-Lagrange formula, we derive that

$$\begin{aligned} \left| {\mathcal {E}}_0(\phi +\psi )-{\mathcal {E}}_0(\phi )-(\nabla \phi ,\nabla \psi )-\int _{\mathcal {M}}f(\phi )\psi dx\right| =\left| \frac{1}{2}\left\| \psi \right\| ^2+\int _0^1\int _{\mathcal {M}}(1-s)f'(\phi +s\psi )\psi dxds\right| . \end{aligned}$$

Owing to the condition (2.37) and the fact that $D(A_1)$ is continuously embedded in $H_2$ and in $L^\infty ({\mathcal {M}})$, we infer that

$$\begin{aligned}&\left| {\mathcal {E}}_0(\phi +\psi )-{\mathcal {E}}_0(\phi )-(\nabla \phi ,\nabla \psi )-\int _{\mathcal {M}}f(\phi )\psi dx\right| \\&\quad \le c\left| A_1\psi \right| ^2_{L^2}+\left\| \psi \right\| ^2_{L^\infty }\sup _{s\in [0,1]}\left| 1-s\right| \int _0^1\left| f'(\phi +s\psi )\right| _{L^1} \\&\quad \le c\left| A_1\psi \right| ^2_{L^2}+c\left| A_1\psi \right| ^2_{L^2}\int _0^1\left| f'(\phi +s\psi )\right| _{L^1} \\&\quad \le c\left| A_1\psi \right| ^2_{L^2}(1+cf\int _{\mathcal {M}}\int _0^1(1+\left| \phi +s\psi \right| ))dxds\\&\quad \le C({\mathcal {M}},f)\left| A_1\psi \right| ^2_{L^2}(1+\left| \phi \right| _{L^2}+\left| \psi \right| _{L^2}). \end{aligned}$$

Therefore,

$$\begin{aligned} \lim _{\psi \rightarrow 0, \psi \ne 0}\frac{\left| {\mathcal {E}}_0(\phi +\psi )-{\mathcal {E}}_0(\phi )-(\nabla \phi ,\nabla \psi )-\int _{\mathcal {M}}f(\phi )\psi dx\right| }{\left| A_1\psi \right| _{L^2}}=0. \end{aligned}$$

This proves that the first Fréchet derivative ${\mathcal {D}} : D(A_1) \rightarrow {\mathcal {L}}(D(A_1); {\mathbb {R}})$ of ${\mathcal {E}}_0$ is given by

$$\begin{aligned} {\mathcal {D}}{\mathcal {E}}_0(\phi )[\psi ]=(\nabla \phi ,\nabla \psi )+\int _{\mathcal {M}}f(\phi )\psi dx,\quad \phi ,\psi \in D(A_1). \end{aligned}$$

(3.3)

Also, it is easy to see that ${\mathcal {D}}{\mathcal {E}}_0$ is Fréchet-differentiable with ${\mathcal {D}}^2{\mathcal {E}}_0 : D(A_1) \rightarrow {\mathcal {L}}(D(A_1); {\mathcal {L}}(D(A_1); {\mathbb {R}}))$ given by

$$\begin{aligned} {\mathcal {D}}^2{\mathcal {E}}_0(\phi )[\psi ,\varphi ]=(\nabla \psi ,\nabla \varphi )-\int _{\mathcal {M}}f'(\phi )\psi \varphi dx,\quad \phi ,\psi ,\varphi \in D(A_1). \end{aligned}$$

(3.4)

Indeed, by direct computation, we can check that

$$\begin{aligned} \left| {\mathcal {D}}{\mathcal {E}}_0(\phi +\psi )[\varphi ]-{\mathcal {D}}{\mathcal {E}}_0(\phi )[\varphi ]-{\mathcal {D}}^2{\mathcal {E}}_0(\phi )[\psi ,\varphi ]\right|&=\left| \int _{\mathcal {M}}(f(\phi +\psi )-f(\phi ))\varphi dx-\int _{\mathcal {M}}f'(\phi )\psi \varphi dx\right| \\&=\left| \int _0^1\int _{\mathcal {M}}(f'(\phi +s\psi )-f'(\phi ))\psi \varphi dx\right| \end{aligned}$$

By the embedding of $D(A_1)$ in $L^\infty ({\mathcal {M}})$ and in $L_0^2({\mathcal {M}})$, the mean value theorem and (2.37), we note that

$$\begin{aligned}&\left| \int _0^1\int _{\mathcal {M}}(f'(\phi +s\psi )-f'(\phi ))\psi \varphi dx\right| \nonumber \\&\quad \le \left| \psi \right| _{L^\infty }\left| \varphi \right| _{L^\infty }\int _0^1\left| f'(\phi +s\psi )-f'(\phi )\right| _{L^1}ds \nonumber \\&\quad \le c\left| A_1\psi \right| _{L^2}\left| A_1\varphi \right| _{L^2}\int _0^1\left| f'(\phi +s\psi )-f'(\phi )\right| _{L^1}ds \nonumber \\&\quad \le c\left| A_1\psi \right| _{L^2}\left| A_1\varphi \right| _{L^2}\int _0^1\left| s\psi f^{''}(\xi ) \right| _{L^1}ds \nonumber \\&\quad \le \frac{c\left| {\mathcal {M}}\right| ^{1/2}}{2}\left| A_1\psi \right| _{L^2}\left| A_1\varphi \right| _{L^2}\left| \psi \right| _{L^2} \nonumber \\&\quad \le \frac{c\left| {\mathcal {M}}\right| ^{1/2}}{2}\left| A_1\psi \right| ^2_{L^2}\left| A_1\varphi \right| _{L^2}. \end{aligned}$$

(3.5)

From (3.5), we arrive at

$$\begin{aligned} \sup _{\left| A_1\varphi \right| _{L^2}\le 1}\left[ \frac{1}{\left| A_1\psi \right| _{L^2}} \left| \int _0^1\int _{\mathcal {M}}(f'(\phi +s\psi )-f'(\phi ))\psi \varphi dx\right| \right] \le C({\mathcal {M}})\left| A_1\psi \right| _{L^2}\rightarrow 0 \text { as } \psi \rightarrow 0\text { in } D(A_1), \end{aligned}$$

from which we get (3.4). Furthermore, from the hypothesis (H1), we can easily check that the derivatives ${\mathcal {D}}{\mathcal {E}}_0$ and ${\mathcal {D}}^2{\mathcal {E}}_0$ are continuous and bounded on bounded subsets of $D(A_1)$. Hence, observing that ${\mathcal {D}}^2{\mathcal {E}}_0(\phi )=\mu$, we can apply Itô’s formula to ${\mathcal {E}}_0(\phi )$ in the classical version of [8] to derive that

$$\begin{aligned} {\mathcal {E}}_0(\phi (t))&+\int _0^t\left| \nabla \mu \right| _{L^2}^2ds+\int _0^t(u\cdot \nabla \phi ,\mu )ds={\mathcal {E}}_0(\phi _0) +\int _0^t(\sigma _2(s,u,\phi ),\mu )dW^2(s)\nonumber \\&+\frac{1}{2}\int _0^t\sum _{j=1}^{+\infty }\int _{\mathcal {M}}\left[ \left| \nabla \sigma _j^2(s,u,\phi )(x)\right| ^2+\left| f'(\phi (s,x))\right| \left| \sigma _j^2(s,u_m,\phi _m)(x)\right| ^2 \right] dxds. \end{aligned}$$

(3.6)

Adding (3.2) with (3.6) and using (2.8) and the fact that $(u\cdot \nabla \phi , f(\phi ))=0$, we obtain (3.1) which ends the proof of the Proposition 3.1. $\square$

Theorem 3.1

We suppose that the Assumptions (H1)–(H4) hold. Moreover, we assume that $\sigma _1(\cdot ,0,0)\in L^p(\Omega ,{\mathcal {F}},{\mathbb {P}};L^2(0,T;{\mathcal {L}}^2(U;H_1)))$ and that $(u_0,\phi _0)\in L^p(\Omega ,{\mathcal {F}},{\mathbb {P}},{\mathbb {H}})$ satisfies ${\mathbb {E}}{\mathcal {E}}^p(u_0,\phi _0)<\infty$, for all $p\ge 2$. Then the system (2.41) has a unique strong solution.

The rest of this section is devoted to the proof of Theorem 3.1. The method relies on Galerkin approximation and deterministic Gronwall’s lemma. For the existence part, instead of the Minty-Browder technique used in [27], we prove the existence and certain uniform estimates for the sequence $(u_m,\phi _m)_m$ of the approximation. Then, as in [37], we use the properties of stopping times and some basic convergence principles from functional analysis to prove the existence of the solution.

3.1 Existence of Strong Solution

Let us consider a finite dimensional Galerkin approximation of the system (2.41). Consider $\{ (w_j,\psi _j),\ j=1,\ldots \}\subset {\mathbb {V}}$ be a orthogonal basis of ${\mathcal {H}}$, where $\{ w_j,\ j=1,\ldots \}$ and $\{ \psi _j,\ j=1,\ldots \}$ are eigenvectors of $A_0$ and $A_1$ respectively given in the previous section. We set for $m\in {\mathbb {N}}$, ${\mathcal {H}}_m={\text {span}}\{ (w_1,\psi _1),\ldots .,(w_m,\psi _m) \}=H_{1m}\times H_{2m}$. We look for $(u_m,\phi _m)\in {\mathcal {H}}_m$ solutions to the ordinary differential equations

$$\begin{aligned} {\left\{ \begin{array}{ll} du_m(t)+ {\mathcal {P}}_m^1[A_{0}u_m+(\beta *A_0u_m)(t)+B_{0}(u_m,u_m)-R_{0}(A_{1}\phi _m,\phi _m)]dt\\ ={\mathcal {P}}_m^1\sigma _1(t,u_m,\phi _m)dW^1_m(t) +\int _{Z}{\mathcal {P}}_m^1\gamma (t,u_m(t^-),\phi _m(t),z){\tilde{\pi }}(dt,dz), \\ d\phi _m(t)+{\mathcal {P}}_m^2[A_{1}\mu _m+B_{1}(u_m,\phi _m)]dt={\mathcal {P}}_m^2\sigma _2(t,u_m,\phi _m)dW^2_m(t),\\ \mu _m={\mathcal {P}}_m^2[A_1\phi _m+f(\phi _m)],\\ (u_m,\phi _m)(0)={\mathcal {P}}_m(u_{0},\phi _{0}), \end{array}\right. } \end{aligned}$$

(3.7)

where ${\mathcal {P}}_m=({\mathcal {P}}_m^1,{\mathcal {P}}_m^2):H_1\times L^2({\mathcal {M}})\rightarrow {\mathcal {H}}_m$ is the orthogonal projection, $W_m^i(t)={\mathcal {P}}_m^iW^i_t$, for $i=1,2$. Since the deterministic terms of (3.7) are locally Lipschitz (see Lemma 2.1), and ${\mathcal {P}}_m^i\sigma _i(.)$, $i=1,2$ and ${\mathcal {P}}_m^1\gamma (.)$ is globally Lipschitz, the system (3.7) has a unique ${\mathcal {H}}_m$-valued càdlàg local strong solution $(u_m,\phi _m)\in L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^\infty (0,T;{\mathcal {H}}_m) )$ with paths $u\in {\mathbf {D}}(0,T;H_{1m})$ and $\phi \in C([0; T];H_{2m} )$, ${\mathbb {P}}$-a.s. (see [1, 27]). Let us now derive the a-priori energy estimates satisfied by the system (3.7).

Proposition 3.2

Let $(u_m,\phi _m)$ be the unique solution of the system (3.7) with $(u_0,\phi _0)\in L^p(\Omega ,{\mathcal {F}},{\mathbb {P}}; {\mathcal {H}})$, for all $p\ge 2$. If $(u_0,\phi _0)$ is such that ${\mathbb {E}}{\mathcal {E}}^p(u_0,\phi _0)<\infty$ for all $p\ge 2$, then there exists a positive constant C independent of m such that for all $p\ge 2$,

$$\begin{aligned}&{\mathbb {E}}\left[ \sup _{0\le t\le T}{\mathcal {E}}^p(u_m,\phi _m)(s)+\left( \int _0^T\left( \left| \nabla \mu _m\right| _{L^2}^2+\left\| u_m\right\| ^2\right) ds\right) ^p \right] \le C(1+{\mathbb {E}}{\mathcal {E}}^p(u_0,\phi _0)), \end{aligned}$$

(3.8)

$$\begin{aligned}&{\mathbb {E}}\sup _{s\in [0,T]}{\mathcal {E}}(u_m,\phi _m)(s)+{\mathbb {E}}\int _0^{T}\left( \left| \nabla \mu _m\right| _{L^2}^2+\left\| u_m\right\| ^2\right) ds\le C(u_0,\phi _0). \end{aligned}$$

(3.9)

Proof

By finite dimensional Itô’s formula (see [4, Theorem 4.4.7]) and the fact that $b_0(u_m,u_m,u_m)=0$, we obtain for all $t \in [0, T ]$,

$$\begin{aligned} \left| u_m(t)\right| ^2_{L^2}&=\left| u_m(0)\right| ^2_{L^2}-2\int _0^t\langle A_{0}u_m+(\beta *A_0u_m)(s)-R_{0}(A_{1}\phi _m,\phi _m), u_m\rangle ds\nonumber \\&\quad +\int _0^t\left| {\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m)\right| ^2_{{\mathcal {L}}^2(U;H_1)}ds+2\int _0^t({\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m),u_m)dW^1_m(s)\nonumber \\&\quad +2\int _0^t\int _{Z}({\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z),u_m){\tilde{\pi }}(ds,dz)+\int _0^t\int _{Z}\Psi _m(s,z)\pi (ds,dz), \end{aligned}$$

(3.10)

where

$$\begin{aligned} \Psi _m(s,z)&=\left| u_m(s^-)+{\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z)\right| ^2_{L^2}-\left| u_m(s^-)\right| ^2_{L^2}\nonumber \\&\quad -2({\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z),u_m). \end{aligned}$$

(3.11)

Note that (2.30) easily gives

$$\begin{aligned} \int _0^t\langle (\beta *A_0u_m) ,u_m\rangle ds =\int _0^t((\beta *\nabla u_m) ,\nabla u_m)ds \ge 0. \end{aligned}$$

(3.12)

Therefore, using the fact that $\left| x\right| ^2_{L^2}-\left| y\right| ^2_{L^2}+\left| x-y\right| ^2_{L^2}=2(x-y,x),$ $\forall x, y\in H_1$, we infer from (3.10) that

$$\begin{aligned} \left| u_m(t)\right| ^2_{L^2}&\le \left| u_m(0)\right| ^2_{L^2}-2\int _0^t\langle A_{0}u_m+(\beta *A_0u_m)(s)-R_{0}(A_{1}\phi _m,\phi _m), u_m\rangle ds\nonumber \\&\quad +2\int _0^t({\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m),u_m)dW^1_m(s)\nonumber \\&\quad +\int _0^t\int _{Z}\left| {\mathcal {P}}_m^1\gamma (s,u_m(s),\phi _m(s),z)\right| _{L^2}^2\pi (ds,dz)+\int _0^t\left| {\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m)\right| ^2_{{\mathcal {L}}^2(U;H_1)}ds\nonumber \\&\quad +2\int _0^t\int _{Z}({\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z),u_m){\tilde{\pi }}(ds,dz). \end{aligned}$$

(3.13)

Note that, since ${\text {spam}}\{ \psi _1,\ldots .\psi _m \}\subset D(A_1)$, we infer that $D{\mathcal {E}}_0(\phi _m)=\mu _m$. Therefore, applying Itô’s formula to the process ${\mathcal {E}}_0(\phi _m)$, we get

$$\begin{aligned}&2{\mathcal {E}}_0(\phi _m(t))+\int _0^t\left| \nabla \mu _m\right| _{L^2}^2ds+\int _0^t(u_m\cdot \nabla \phi _m,\mu _m)ds\nonumber \\&\quad =2{\mathcal {E}}_0(\phi _m(0)) +2\int _0^t({\mathcal {P}}_m^2\sigma _2(s,u_m,\phi _m),\mu _m)dW^2_m(s)\nonumber \\&\qquad +\int _0^t\sum _{j=1}^m\int _{\mathcal {M}}\left[ \left| \nabla {\mathcal {P}}_m^2\sigma _j^2(s,u_m,\phi _m)(x)\right| ^2+\left| f'(\phi _m(s,x))\right| \left| {\mathcal {P}}_m^2\sigma _j^2(s,u_m,\phi _m)(x)\right| ^2 \right] dxds. \end{aligned}$$

(3.14)

Using the fact that $H_1\hookrightarrow L^p({\mathcal {M}})$, $p\ge 2$, by assumption (H1), we get

$$\begin{aligned}&\int _0^t\sum _{j=1}^m\int _{\mathcal {M}}\left[ \left| \nabla {\mathcal {P}}_m^2\sigma _j^2(s,u_m,\phi _m)(x)\right| ^2+\left| f'(\phi _m(s,x))\right| \left| {\mathcal {P}}_m^2\sigma _j^2(s,u_m,\phi _m)(x)\right| ^2 \right] dxds\nonumber \\&\quad \le c\int _0^t\left\| \sigma _2(s,u_m,\phi _m)\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds+C_f\sum _{j=1}^m\int _0^t\int _{\mathcal {M}}\left| \phi _m\right| \left| {\mathcal {P}}_m^2\sigma _j^2(s,u_m,\phi _m)(x)\right| ^2dxds\nonumber \\&\quad \le c\int _0^t\left\| \sigma _2(s,u_m,\phi _m)\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds+C_f\sum _{j=1}^m\int _0^t\int _{\mathcal {M}}\left| \phi _m\right| \left| {\mathcal {P}}_m^2\sigma _j^2(s,u_m,\phi _m)\right| ^2(x)dxds\nonumber \\&\quad \le c\int _0^t\left\| \sigma _2(s,u_m,\phi _m)\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds+C_f\sum _{j=1}^m\int _0^t\left\| \phi _m\right\| _{L^2}^2\left\| {\mathcal {P}}_m^2\sigma _j^2(s,u_m,\phi _m)\right\| _{L^4}^2ds\nonumber \\&\quad \le c\int _0^t\left\| \sigma _2(s,u_m,\phi _m)\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds+c\int _0^t\left\| \phi _m\right\| ^2\left\| \sigma _2(s,u_m,\phi _m)\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds. \end{aligned}$$

(3.15)

Now, for each $n\ge 1,$ we consider the ${\mathcal {F}}_t$-stopping time $\tau _n^m$ defined by:

$$\begin{aligned} \tau _n^m=\min \left( T,\inf \left\{ t\in [0,T]: {\mathcal {E}}(u_m,\phi _m)(t)+ 2\int _0^t\left( \left\| u_m\right\| ^2+\left\| \mu _m\right\| ^2\right) ds\ge n^2\right\} \right) . \end{aligned}$$

(3.16)

For fixed m,  the sequence $\{\tau _n^m;n\ge 1 \}$ is increasing to T. Adding (3.10) with (3.14) after using (3.12) and (3.15), we get for all $t\in [0,T]$,

$$\begin{aligned}&{\mathcal {E}}(u_m,\phi _m)(t\wedge \tau _n^m) \nonumber \\&\quad +2\int _0^{t\wedge \tau _n^m}\left( \left| \nabla \mu _m\right| _{L^2}^2+\left\| u_m\right\| ^2\right) ds\le {\mathcal {E}}(u_0,\phi _0)\nonumber \\&\quad +\int _0^{t\wedge \tau _n^m}\left| {\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m)\right| ^2_{{\mathcal {L}}^2(U;H_1)}ds +2\int _0^{t\wedge \tau _n^m}({\mathcal {P}}_m^2\sigma _2(s,u_m,\phi _m),\mu _m)dW^2_m(s)\nonumber \\&\quad +c\int _0^{t\wedge \tau _n^m}\left\| \sigma _2(s,u_m,\phi _m)\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds+c\int _0^t\left\| \phi _m\right\| ^2\left\| \sigma _2(s,u_m,\phi _m)\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds\nonumber \\&\quad +2\int _0^{t\wedge \tau _n^m}({\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m),u_m)dW^1_m(s)+\int _0^t\int _{Z}\Psi _m(s,z)\pi (ds,dz)\nonumber \\&\quad +2\int _0^{t\wedge \tau _n^m}\int _{Z}({\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z),u_m){\tilde{\pi }}(ds,dz). \end{aligned}$$

(3.17)

Now raising both sides to the power $p\ge 2$, taking supremum over $s \in [0, t\wedge \tau _n^m]$ and taking mathematical expectation we have

$$\begin{aligned}&{\mathbb {E}}\sup _{r\in [0,t\wedge \tau _n^m]}{\mathcal {E}}^p(u_m,\phi _m)(r) \nonumber \\&\quad +4{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| \nabla \mu _m\right| _{L^2}^2\right) ^pds+ 4{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left\| u_m\right\| ^2\right) ^pds\le {\mathcal {E}}^p(u_0,\phi _0)\nonumber \\&\quad +{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| {\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m)\right| ^2_{{\mathcal {L}}^2(U;H_1)}ds\right) ^p+c{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}(1+\left\| \phi _m\right\| ^2)ds\right) ^p\nonumber \\&\quad +{\mathbb {E}}\sup _{r\in [0,t\wedge \tau _n^m]}\left| \int _0^r({\mathcal {P}}_m^2\sigma _2(s,u_m,\phi _m),\mu _m)dW^2_m(r)\right| ^p\nonumber \\&\quad +2{\mathbb {E}}\sup _{r\in [0,t\wedge \tau _n^m]}\left| \int _0^r({\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m),u_m)dW^1_m(s)\right| ^p\nonumber \\&\quad +c{\mathbb {E}}\sup _{0\le r\le t\wedge \tau _n^m}\left| I_5(r)\right| ^p+c{\mathbb {E}}\sup _{0\le r\le t\wedge \tau _n^m}\left| I_6(r)\right| ^p. \end{aligned}$$

(3.18)

where

$$\begin{aligned} I_5(r)&=\int _0^{r}\int _{Z}\left\{ \left| u_m(\tau ^-)+{\mathcal {P}}_m^1\gamma (\tau ,u_m(\tau ^-),\phi _m(\tau ),z)\right| _{L^2}^2-\left| u_m(\tau ^-)\right| _{L^2}^2 \right\} {\tilde{\pi }}(d\tau ,dz), \nonumber \\ I_6(r)&=\int _0^{r}\int _{Z} \left\{ \left| u_m(s)+{\mathcal {P}}_m^1\gamma (s,u_m(s),\phi _m(s),z)\right| _{L^2}^2-\left| u_m(s)\right| _{L^2}^2 \right\} \lambda (dz)ds\nonumber \\&\quad -2\int _0^{r}\int _{Z} ({\mathcal {P}}_m^1\gamma (s,u_m(s),\phi _m(s),z),u_m(s)\lambda (dz)ds\nonumber \\&\le c\int _0^{r}\int _{Z}\left| {\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z)\right| _{L^2}^2\lambda (dz)ds\nonumber \\&\le cK_0\int _0^{r}(1+\left| (u_m,\phi _m)\right| _{\mathcal {H}}^2)ds. \end{aligned}$$

(3.19)

As in [6, 7], we note that

$$\begin{aligned}&\int _{Z}\left\{ \left| u_m(s^-)+{\mathcal {P}}_m^1\gamma (s,u_m(s-),\phi _m(s),z)\right| _{L^2}^2-\left| u_m(\tau ^-)\right| _{L^2}^2 \right\} ^2\lambda (dz)\nonumber \\&\quad \le \left| u_m(s^-)\right| ^2_{L^2}\int _{Z}\left| {\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z)\right| _{L^2}^2\lambda (dz)\nonumber \\&\qquad +c\int _{Z}\left| {\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z)\right| _{L^2}^4\lambda (dz)\nonumber \\&\quad \le c_0+c_1\left| (u_m,\phi _m)\right| _{\mathcal {H}}^2+c_2\left| (u_m,\phi _m)\right| _{\mathcal {H}}^4\nonumber \\&\quad \le K_2+K_3\left| (u_m,\phi _m)\right| _{\mathcal {H}}^4. \end{aligned}$$

(3.20)

It follows that

$$\begin{aligned}&\left( \int _0^{t\wedge \tau _n^m}\int _{Z}\left\{ \left| u_m(s^-)+{\mathcal {P}}_m^1\gamma (s,u_m(s-),\phi _m(s),z)\right| _{L^2}^2-\left| u_m(\tau ^-)\right| _{L^2}^2 \right\} ^2\lambda (dz)ds\right) ^{p/2}\nonumber \\&\quad \le c(K_2T)^{p/2}+c(K_3)^{p/2}\left( \int _0^{t\wedge \tau _n^m}{\mathcal {E}}^2(u_m,\phi _m)ds\right) ^{p/2}. \end{aligned}$$

(3.21)

Applying Burkholder–Davis–Gundy’s inequality (see [33, Theorem 48]) and using (3.20)–(3.21), we derive that

$$\begin{aligned} {\mathbb {E}}\sup _{0\le r\le t\wedge \tau _n^m}\left| I_5(r)\right| ^p&\le \frac{1}{4}{\mathbb {E}}\sup _{r\in [0,t\wedge \tau _n^m]}{\mathcal {E}}^p(u_m,\phi _m)(r)+c(K_2T)^{p/2}+c(K_3)^{p/2}{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}{\mathcal {E}}(u_m,\phi _m)ds\right) ^{p}\nonumber \\&\le \frac{1}{4}{\mathbb {E}}\sup _{r\in [0,t\wedge \tau _n^m]}{\mathcal {E}}^p(u_m,\phi _m)(r)+c(K_2T)^{p/2}+c(K_3)^{p/2}T^{p-1}{\mathbb {E}}\int _0^{t\wedge \tau _n^m}{\mathcal {E}}^p(u_m,\phi _m)ds. \end{aligned}$$

(3.22)

Using Hölder’s inequality, it follows from (3.19) that

$$\begin{aligned} {\mathbb {E}}\left| I_6(t\wedge \tau _n^m)\right| ^p&\le cK_0^p{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}(1+\left| (u_m,\phi _m)\right| _{\mathcal {H}}^2)ds\right) ^p\nonumber \\&\le C(K_0,T)+cK_0^2\int _0^{t\wedge \tau _n^m}{\mathbb {E}}{\mathcal {E}}^p(u_m,\phi _m)(s)ds. \end{aligned}$$

(3.23)

By Doob’s inequality, we derive will the help of (2.38)$_1$ and Hölder’s inequality that

$$\begin{aligned}&2{\mathbb {E}}\sup _{r\in [0,t\wedge \tau ^m_n]}\left| \int _0^r({\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m),u_m)dW^1_m(r)\right| ^p\nonumber \\&\quad \le c{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| {\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m)\right| _{{\mathcal {L}}^2(U;H_1)}^2\left| u_m\right| ^2_{L^2}dr\right) ^{p/2} \nonumber \\&\quad \le c{\mathbb {E}}\left[ \sup _{0\le r\le t\wedge \tau _n^m}{\mathcal {E}}(u_m,\phi _m)(r)\int _0^t\left\| {\mathcal {P}}_m^i\sigma _i(s,u_m,\phi _m)\right\| ^2_{{\mathcal {L}}^2(U;H_1)}ds \right] ^{p/2}\nonumber \\&\quad \le \frac{1}{4}{\mathbb {E}}\sup _{0\le r\le t\wedge \tau _n^m}{\mathcal {E}}^p(u_m,\phi _m)(r)+c{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left\| {\mathcal {P}}_m^i\sigma _i(s,u_m,\phi _m)\right\| ^2_{{\mathcal {L}}^2(U;H_1)}dr\right) ^p\nonumber \\&\quad \le \frac{1}{4}{\mathbb {E}}\sup _{0\le r\le t\wedge \tau _n^m}{\mathcal {E}}^p(u_m,\phi _m)(r)+C(K_0,T)+cK_0^p\int _0^{t\wedge \tau _n^m}{\mathbb {E}}{\mathcal {E}}^p(u_m,\phi _m)(s)dr. \end{aligned}$$

(3.24)

By Hölder’s inequality, we derive that

$$\begin{aligned}&{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| {\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m)\right| ^2_{{\mathcal {L}}^2(U;H_1)}ds\right) ^p+c{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}(1+\left\| \phi _m\right\| ^2)ds\right) ^p\nonumber \\&\quad \le C(T,L){\mathbb {E}}\int _0^{t\wedge \tau _n^m}(1+{\mathcal {E}}^p(u_m,\phi _m))ds+c{\mathbb {E}}\left( \int _0^T\left| \sigma _1(s,0,0)\right| ^2_{{\mathcal {L}}^2(U;H_1)}ds\right) ^p\nonumber \\&\quad \le C(T,L)+C(T,L){\mathbb {E}}\int _0^{t\wedge \tau _n^m}{\mathcal {E}}^2(u_m,\phi _m)(s)ds+c{\mathbb {E}}\left( \int _0^T\left| \sigma _1(s,0,0)\right| ^2_{{\mathcal {L}}^2(U;H_1)}ds\right) ^p. \end{aligned}$$

(3.25)

By (2.37), (3.7)$_4$, and the fact that $H_1\hookrightarrow L^2({\mathcal {M}})$ we infer that

$$\begin{aligned} \left| \langle \mu _m\rangle \right| \le \left| {\mathcal {M}}\right| ^{-1}\left\| f(\phi _m)\right\| _{L^1}\le \left| {\mathcal {M}}\right| ^{-1}C_f(1+\left| \phi _m\right| ^2_{L^2})\le c\left| {\mathcal {M}}\right| ^{-1}(1+\left\| \phi _m\right\| ^2). \end{aligned}$$

(3.26)

By Doob’s inequality, Hölder’s inequality, the condition (2.29), (3.26) and Young’s and Poincaré-Wirtinger’s inequalities, we infer that

$$\begin{aligned}&{\mathbb {E}}\sup _{r\in [0,t\wedge \tau _n^m]}\left| \int _0^r({\mathcal {P}}_m^2\sigma _2(s,u_m,\phi _m),\mu _m)dW^2_m(r)\right| ^p\nonumber \\&\quad \le c{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| {\mathcal {P}}_m^1\sigma _1(s,u_m,\phi _m)\right| ^2_{{\mathcal {L}}^2(U;L^2(D))}\left| \mu _m\right| ^2_{L^2}ds\right) ^{p/2}\nonumber \\&\quad \le cK_0^{p/2}{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| \nabla \mu _m\right| ^2_{L^2}ds+\int _0^{t\wedge \tau _n^m}\left| \langle \mu _m\rangle \right| ^2ds\right) ^{p/2}\nonumber \\&\quad \le cK_0^{p/2}{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| \nabla \mu _m\right| ^2_{L^2}ds\right) ^{p/2}+ cK_0^{p/2}{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| \langle \mu _m\rangle \right| ^2ds\right) ^{p/2}\nonumber \\&\quad \le cK_0^{p/2}\left[ {\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| \nabla \mu _m\right| ^2_{L^2}ds\right) ^p\right] ^{1/2}+C(T)K_0^{p/2}{\mathbb {E}}\int _0^{t\wedge \tau _n^m}(1+\left\| \phi _m\right\| ^{2p})ds\nonumber \\&\quad \le \frac{cK_0^{p}}{2}+\frac{1}{2}{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| \nabla \mu _m\right| ^2_{L^2}ds\right) ^p+C(T)K_0^{p/2}{\mathbb {E}}\int _0^{t\wedge \tau _n^m}(1+\left\| \phi _m\right\| ^{2p})ds\nonumber \\&\quad \le \frac{cK_0^{p}}{2}+\frac{1}{2}{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| \nabla \mu _m\right| ^2_{L^2}ds\right) ^p+C(T)K_0^{p/2}{\mathbb {E}}\int _0^{t\wedge \tau _n^m}{\mathcal {E}}^p(u_m,\phi _m)(s)ds. \end{aligned}$$

(3.27)

It follows from (3.18)–(3.27) that

$$\begin{aligned}&{\mathbb {E}}\sup _{r\in [0,t\wedge \tau _n^m]}{\mathcal {E}}^p(u_m,\phi _m)(r) \nonumber \\&\quad +3{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left| \nabla \mu _m\right| _{L^2}^2ds\right) ^p+ 4{\mathbb {E}}\left( \int _0^{t\wedge \tau _n^m}\left\| u_m\right\| ^2ds\right) ^p\le {\mathcal {E}}^p(u_0,\phi _0)+C(T,K_0)\nonumber \\&\quad +C(T,K_0,L)\int _0^{t\wedge \tau _n^m}{\mathbb {E}}{\mathcal {E}}^p(u_m,\phi _m)(s)ds+c{\mathbb {E}}\left( \int _0^T\left| \sigma _1(s,0,0)\right| ^2_{{\mathcal {L}}^2(U;H_1)}ds\right) ^p, \end{aligned}$$

(3.28)

(3.8) follows from Gronwall’s lemma and the fact that $\tau _n^m\nearrow T$ as n goes to $\infty$.

By Young’s inequality, and (3.8), we derive that

$$\begin{aligned} {\mathbb {E}}\sup _{s\in [0,T]}{\mathcal {E}}(u_m,\phi _m)(s)&\le \frac{1}{2}{\mathbb {E}}\sup _{s\in [0,T]}{\mathcal {E}}^2(u_m,\phi _m)(s)+\frac{1}{2}\le C(1+{\mathbb {E}}{\mathcal {E}}^2(u_0,\phi _0)), \end{aligned}$$

(3.29)

$$\begin{aligned} {\mathbb {E}}\int _0^{T}\left( \left| \nabla \mu _m\right| _{L^2}^2+\left\| u_m\right\| ^2\right) ds&\le \frac{1}{2}{\mathbb {E}}\left( \int _0^{T}\left( \left| \nabla \mu _m\right| _{L^2}^2+\left\| u_m\right\| ^2\right) ds\right) ^2+\frac{1}{2}\nonumber \\&\le C(1+{\mathbb {E}}{\mathcal {E}}^2(u_0,\phi _0)). \end{aligned}$$

(3.30)

From (3.29) and (3.30) we get (3.9). The Proposition 3.2 is then proved. $\square$

Corollary 3.1

Under the same hypothesis as in Proposition 3.2, there exists a positive constant C independent of m such that for all $p\ge 2$,

$$\begin{aligned}&{\mathbb {E}}\left[ \sup _{0\le t\le T}\left| (u_m,\phi _m)(t)\right| _{{\mathcal {H}}}^2+\int _0^T\left( \left\| (u_m,\phi _m)(s)\right\| _{\mathbb {V}}^2+\left| A_1^{3/2}\phi _m\right| ^2_{L^2}\right) ds \right] \le C, \end{aligned}$$

(3.31)

$$\begin{aligned}&{\mathbb {E}}\left[ \sup _{0\le t\le T}\left| (u_m,\phi _m)(t)\right| _{{\mathcal {H}}}^p+\left( \int _0^T\left( \left\| (u_m,\phi _m)(s)\right\| _{\mathbb {V}}^2+\left| A_1^{3/2}\phi _m\right| ^2_{L^2}\right) ds\right) ^p \right] \le C, \end{aligned}$$

(3.32)

$$\begin{aligned}&{\mathbb {E}}\int _0^T\left| f(\phi _m)\right| ^2_{L^2}ds+{\mathbb {E}}\int _0^T\left\| B_1(u_m,\phi _m)(s)\right\| ^2_{H_2^*}ds\le C, \end{aligned}$$

(3.33)

$$\begin{aligned}&{\mathbb {E}}\int _0^T\left\| B_0(u_m,u_m)(s)\right\| ^2_{V_1^*}ds+{\mathbb {E}}\int _0^T\left\| R_0(A_1\phi _m,\phi _m)(s)\right\| ^2_{V_1^*}ds\le C, \end{aligned}$$

(3.34)

$$\begin{aligned}&\quad \sum _{i=1}^2{\mathbb {E}}\left[ \int _0^T\left| {\mathcal {P}}_m^i\sigma _i(s,u_m,\phi _m)\right| ^2_{{\mathcal {L}}^2(U;H_i)}ds\right] \nonumber \\&\qquad +{\mathbb {E}}\left[ \int _0^T\int _{Z_m}\left| {\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z)\right| _{L^2}^2\lambda (dz)ds\right] \le C. \end{aligned}$$

(3.35)

Proof

By (3.7)$_4$, (2.37) and the Poincaré–Wirtinger inequality, we note that

$$\begin{aligned} \left| f(\phi _m)\right| ^2_{L^2}&\le C_f(1+\left\| \phi _m\right\| ^4)\le C_f(1+{\mathcal {E}}^2(u_m,\phi _m)), \end{aligned}$$

(3.36)

$$\begin{aligned} \left| A_1\phi _m\right| ^2_{L^2}&\le c\left| \mu _m\right| ^2_{L^2}+c\left| f(\phi _m)\right| ^2_{L^2}\nonumber \\&\le c\left| \nabla \mu _m\right| ^2_{L^2}+c\left| \langle \mu _m\rangle \right| ^2+c\left| f(\phi _m)\right| ^2_{L^2}\\&\le c\left| \nabla \mu _m\right| ^2_{L^2}+c\left| {\mathcal {M}}\right| ^{-2}C_f(1+\left\| \phi _m\right\| ^2)+c\left| f(\phi _m)\right| ^2_{L^2}\nonumber \\&\le c\left| \nabla \mu _m\right| ^2_{L^2}+C({\mathcal {M}},f)(1+{\mathcal {E}}^2(u_m,\phi _m)),\nonumber \end{aligned}$$

(3.37)

$$\begin{aligned} \left| A_1^{3/2}\phi _m\right| ^2_{L^2}&\le c\left| \nabla \mu _m\right| ^2_{L^2}+c\left| f'(\phi _m)\nabla \phi _m\right| ^2_{L^2}\nonumber \\&\le c\left| \nabla \mu _m\right| ^2_{L^2}+C_f(\left\| \phi _m\right\| ^2\left| A_1\phi _m\right| ^2_{L^2}+\left| A_1\phi _m\right| ^2_{L^2}). \end{aligned}$$

(3.38)

The estimates (3.31), (3.32) and the first part of (3.33) follows from (3.36)–(3.38) and the Proposition 3.2. By (2.10), (2.11) and (2.12), we also note that

$$\begin{aligned}&{\mathbb {E}}\int _0^T\left\| B_0(u_m,u_m)(s)\right\| ^2_{V_1^*}ds+{\mathbb {E}}\int _0^T\left\| R_0(A_1\phi _m,\phi _m)(s)\right\| ^2_{V_1^*}ds\nonumber \\&\quad \le c{\mathbb {E}}\int _0^T\left| u_m(s)\right| ^2_{L^2}\left\| u_m(s)\right\| ^2ds+{\mathbb {E}}\int _0^T\left\| \phi _m(s)\right\| ^2\left| A_1^{3/2}\phi _m\right| ^2_{L^2}ds\nonumber \\&\quad \le \left( {\mathbb {E}}\sup _{0\le s\le T}\left| u_m(s)\right| _{L^2}^4\right) ^{1/2}\left[ {\mathbb {E}}\left( \int _0^T\left\| u_m\right\| ^2ds\right) ^2 \right] ^{1/2}\nonumber \\&\qquad +c\left( {\mathbb {E}}\sup _{0\le s\le T}\left\| \phi _m(s)\right\| ^4\right) ^{1/2}\left[ {\mathbb {E}}\left( \int _0^T\left| A_1^{3/2}\phi _m\right| _{L^2}^2ds\right) ^2 \right] ^{1/2}\le C(u_0,\phi _0)<\infty , \end{aligned}$$

(3.39)

$$\begin{aligned} {\mathbb {E}}\int _0^T\left\| B_1(u_m,\phi _m)(s)\right\| ^2_{H_2^*}ds&\le c{\mathbb {E}}\int _0^T\left| u_m\right| ^2_{L^2}\left\| \phi _m\right\| \left| A_1\phi _m\right| _{L^2}ds\nonumber \\&\le c{\mathbb {E}}\int _0^T\left| u_m\right| ^4_{L^2}ds+{\mathbb {E}}\int _0^T\left\| \phi _m\right\| ^2\left| A_1\phi _m\right| _{L^2}^2ds\le C, \end{aligned}$$

(3.40)

$$\begin{aligned}&\sum _{i=1}^2{\mathbb {E}}\left[ \int _0^T\left| {\mathcal {P}}_m^i\sigma _i(s,u_m,\phi _m)\right| ^2_{{\mathcal {L}}^2(U;H_i)}ds\right] \nonumber \\&\qquad +{\mathbb {E}}\left[ \int _0^T\int _{Z}\left| {\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z)\right| _{L^2}^2\lambda (dz)ds\right] \nonumber \\&\quad \le K_1T+K_1{\mathbb {E}}\int _0^T\left| (u_m,\phi _m)(s)\right| _{\mathcal {H}}^2ds+\sum _{i=1}^2{\mathbb {E}}\left[ \int _0^T\left| \sigma _i(s,0,0)\right| ^2_{{\mathcal {L}}^2(U;H_i)}ds\right] \nonumber \\&\quad \le K_1T+K_1T{\mathbb {E}}\sup _{0\le s\le T}{\mathcal {E}}(u_m,\phi _m)(s)\le C(u_0,\phi _0)<\infty . \end{aligned}$$

(3.41)

By (3.39)–(3.41) we end the proof of Corollary 3.1. $\square$

From the Corollary 3.1, and along with the Banach-Alaoglu theorem, one can extract a subsequence still denoted by $(u_m,\phi _m)$ to simplify the notation which converges to the following limits

$$\begin{aligned} \begin{array}{lll} (u_m,\phi _m){\mathop {\rightharpoonup }\limits ^{*}}(u,\phi )\qquad \text {in}\qquad L^p(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{\infty }(0,T;{\mathcal {H}})),\\ (u_m,\phi _m)\rightharpoonup (u,\phi )\qquad \text {in}\qquad L^p(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;{\mathbb {V}})),\\ (u_m,\phi _m)\rightharpoonup (u,\phi )\qquad \text {in}\qquad L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;{\mathbb {V}})),\\ \phi _m\rightharpoonup \phi \qquad \text {in}\qquad L^p(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;D(A_1^{3/2}))),\\ B_0(u_m,\phi _m)\rightharpoonup B_0^0\qquad \text {in}\qquad L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;V_1^*)),\\ R_0(A_1\phi _m,\phi _m)\rightharpoonup R_0^0\qquad \text {in}\qquad L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;V_1^*)),\\ B_1(u_m,\phi _m)\rightharpoonup B_1^0\qquad \text {in}\qquad L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;H_2^*),\\ f(\phi _m)\rightharpoonup f^0\qquad \text {in}\qquad L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;L^2({\mathcal {M}})),\\ {\mathcal {P}}_m^i\sigma _i(\cdot ,u_m,\phi _m)\longrightarrow \varPhi _i(\cdot )\qquad \text {in}\qquad L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;{\mathcal {L}}^2(U;H_i))),\quad i=1,2,\\ {\mathcal {P}}_m^1\gamma (\cdot ,u_m(\cdot ),\phi _m(\cdot ))\rightharpoonup \Psi (\cdot )\qquad \text {in}\qquad {\mathcal {M}}^2_T(H_1). \end{array} \end{aligned}$$

(3.42)

With these convergence at hand we see from (3.7) that $(u,\phi )(.)$ satisfies the following Itô stochastic differential: For all $t\in [0,T]$,

$$\begin{aligned}&u(t)+\int _0^t(\beta *A_0u)ds+\int _0^tA_0uds+\int _0^tB_0^0(s)ds=u_0+\int _0^tR_0^0(s)ds+\int _0^t\varPhi _1(s)dW^1_s\nonumber \\&\qquad +\int _0^t\int _Z\Psi (s,z){\tilde{\pi }}(ds,dz),\nonumber \\&\quad \qquad \phi (t)+\int _0^tA_1\mu ^0 ds+\int _0^tB_1^0(s)ds=\phi _0+\int _0^t\varPhi _2(s)dW^2_s,\nonumber \\&\quad \qquad \mu ^0=A_1\phi +f^0, \end{aligned}$$

(3.43)

${\mathbb {P}}$-almost surely as equality in $V_1^*\times H_2^*$.

From the energy estimates (see Corollary 3.1), $(u_m,\phi _m)$ is almost surely uniformly convergent on finite intervals [0, T] to $(u,\phi )$, from which it follows that $(u,\phi )$ is ${\mathcal {F}}_t$-adapted and the ${\mathcal {F}}_t$-adapted paths of u are càdlàg while the ${\mathcal {F}}_t$-adapted paths of $\phi$ are continuous (see [4, Theorem 6.2.3]).

Proposition 3.3

We have the following identities

$$\begin{aligned} \begin{aligned}&B_0(u,u)=B_0^0,\quad R_0(A_1\phi ,\phi )=R_0^0,\quad B_1(u,\phi )=B_1^0, \\&f(\phi _m)=f^0,\quad \gamma (s,u,\phi )= \Psi (s),\quad \sigma _i(s,u,\phi )= \varPhi _i(s),\ i=1,2. \end{aligned} \end{aligned}$$

(3.44)

Proof

Let $({\tilde{u}}_m,{\tilde{\phi }}_m,{\tilde{\mu }}_m)={\mathcal {P}}^0_m(u,\phi ,\mu ),$ where ${\mathcal {P}}^0_m=({\mathcal {P}}^1_m,{\mathcal {P}}^2_m, {\mathcal {P}}^2_m)$. We have

$$\begin{aligned} \begin{aligned}&\left| ({\tilde{u}}_m,{\tilde{\phi }}_m)\right| _{\mathcal {H}}\le \left| (u,\phi )\right| _{\mathcal {H}},\\&\left\| ({\tilde{u}}_m,{\tilde{\phi }}_m)\right\| _{\mathbb {V}}\le c\left\| (u,\phi )\right\| _{\mathbb {V}},\\&({\tilde{u}}_m,{\tilde{\phi }}_m)\longrightarrow (u,\phi )\text { in } {\mathbb {V}} \text { for almost every } (\omega ,t)\in \Omega \times [0,T],\\&({\tilde{u}}_m,{\tilde{\phi }}_m)\longrightarrow (u,\phi )\text { in } L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;{\mathbb {V}})). \end{aligned} \end{aligned}$$

(3.45)

From (3.7) and (3.43), we derive that for $1\le k\le m$,

$$\begin{aligned}&\langle {\tilde{u}}_m-u_m(t),w_k\rangle \nonumber \\&\quad +\int _0^t\langle A_0({\tilde{u}}_m-u_m),w_k\rangle ds+\int _0^t\langle (\beta *A_0({\tilde{u}}_m-u_m)),w_k\rangle ds\nonumber \\&\quad +\int _0^t\langle B_0^0(s)-B_0(u_m,u_m),w_k\rangle =\int _0^t\langle R_0^0-R_0(A_1\phi _m,\phi _m),w_k\rangle ds\nonumber \\&\quad +\sum _{j=1}^{m}\int _0^t\langle \varPhi _1(s)e_j-\sigma _1(s,u_m,\phi _m)e_j,w_k\rangle d\beta ^j_s+\sum _{j=m+1}^{+\infty }\int _0^t\langle \varPhi _1(s)e_j,w_k\rangle d\beta ^j_s\nonumber \\&\quad +\int _0^t\int _Z(\Psi (s,z)-\gamma (s,u_m(s^-),\phi _m(s),z),w_k){\tilde{\pi }}(ds,dz) \end{aligned}$$

(3.46)

$$\begin{aligned}&\langle {\tilde{\phi }}_m(t)-\phi (t)_m,\psi _k\rangle \nonumber \\&\quad +\int _0^t\langle A_1({\tilde{\mu }}_m-\mu _m),\psi _k\rangle ds+\int _0^t\langle B_1^0(s)-B_1(u_m,\phi _m),\psi _k\rangle ds=0,\nonumber \\&\quad +\sum _{j=1}^{m}\int _0^t\langle \varPhi _2(s)e_j-\sigma _2(s,u_m,\phi _m)e_j,w_k\rangle d\beta ^j_s+\sum _{j=m+1}^{+\infty }\int _0^t\langle \varPhi _2(s)e_j,w_k\rangle d\beta ^j_s\nonumber \\&\qquad \langle {\tilde{\mu }}_m-\mu _m,A_1\psi _k\rangle =\langle A_1({\tilde{\phi }}_m-\phi _m),A_1\psi _k\rangle +\langle f^0-f(\phi _m),A_1\psi _k\rangle . \end{aligned}$$

(3.47)

Note that since $B_0,$ $R_0$ and $B_1$ are bilinear, we derive that

$$\begin{aligned} B_0^0-B_0(u_m,u_m)= & {} B_0^0-B_0({\tilde{u}}_m,{\widetilde{u}}_m)+B_0({\tilde{u}}_m-u_m,{\tilde{u}}_m)+B_0(u_m,{\tilde{u}}_m-u_m),\nonumber \\ R^0_0-R_0(A_1\phi _m,\phi _m)= & {} R_0^0-R_0(A_1{\tilde{\phi }}_m,{\tilde{\phi }}_m)+R_0(A_1({\tilde{\phi }}_m-\phi _m),{\tilde{\phi }}_m)+R_0(A_1\phi _m,{\tilde{\phi }}_m-\phi _m)\nonumber \\ B_1^0-B_1(u_m,\phi _m)= & {} B_1^0-B_1({\tilde{u}}_m,{\tilde{\phi }}_m)+B_1({\tilde{u}}_m-u_m,{\tilde{\phi }}_m)+B_1(u_m,{\tilde{\phi }}_m-\phi _m),\nonumber \\ f^0-f(\phi _m)= & {} f^0-f({\tilde{\phi }}_m)+f({\tilde{\phi }}_m)-f(\phi _m). \end{aligned}$$

(3.48)

Let us set $\theta _m={\tilde{u}}_m-u_m,$ $\rho _m={\tilde{\phi }}_m-\phi _m,$ $\zeta _m={\tilde{\mu }}_m-\mu _m.$ From Itô’s formula, we have

$$\begin{aligned} d\langle \theta _m,w_k\rangle ^2=2\langle \theta _m,w_k\rangle d\langle \theta _m,w_k\rangle&+\sum _{j=1}^{m}[\langle \varPhi _1(t)e_j-\sigma _1(t,u_m,\phi _m)e_j,w_k\rangle ]^2dt\nonumber \\&+\sum _{j=m+1}^{+\infty }[\langle \varPhi _1(t)e_j,w_k\rangle ]^2dt+\int _{Z}\Upsilon (s,z)\pi (dt,dz), \end{aligned}$$

(3.49)

where

$$\begin{aligned} \Upsilon (s,z)&=\left| u_m(s^-)+{\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z)-{\mathcal {P}}_m^1\varPhi _1(s,z)\right| ^2_{L^2}\\&\quad -\left| u_m(s^-)\right| ^2_{L^2}-2({\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z),u_m)\\&=\left| {\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z)-{\mathcal {P}}_m^1\varPhi _1(s,z)\right| ^2_{L^2}. \end{aligned}$$

It follows that

$$\begin{aligned} \left| \theta _m(t)\right| ^2_{L^2}&+2\int _0^t(\left\| \theta _m\right\| ^2+\langle B_0^0-B_0(u_m,u_m),\theta _m\rangle )ds=2\int _0^t\langle R_0^0-R_0(A_1\phi _m,\phi _m),\theta _m\rangle ds\nonumber \\&+2\sum _{j=1}^{m}\int _0^t\langle \varPhi _1(s)e_j-\sigma _1(s,u_m,\phi _m)e_j,\theta _m\rangle d\beta ^j_s+2\sum _{j=m+1}^{+\infty }\int _0^t\langle \varPhi _1(s)e_j,\theta _m\rangle d\beta ^j_s \nonumber \\&+\sum _{j=1}^{m}\int _0^t\left| {\mathcal {P}}_m^1[ \varPhi _1(s)e_j-\sigma _1(s,u_m,\phi _m)e_j]\right| _{L^2}^2ds+\sum _{j=m+1}^{+\infty }\int _0^t\left| {\mathcal {P}}_m^1 \varPhi _1(s)e_j\right| _{L^2}^2ds\nonumber \\&+2\int _0^t\int _Z({\mathcal {P}}_m^1(\Psi (s,z)-\gamma (s,u_m(s^-),\phi _m(s),z)),\theta _m){\tilde{\pi }}(ds,dz)\nonumber \\&+\int _0^t\int _{Z}\Upsilon (s,z)\pi (ds,dz)-2\int _0^t((\beta *\nabla \theta _m) ,\nabla \theta _m)ds. \end{aligned}$$

(3.50)

Also, applying the Itô formula to the process $\left\| \rho _m\right\| ^2$, and replacing $\psi _k$ in (3.47)$_3$ by ${\overline{\zeta }}_m-\xi \rho _m$, we obtain

$$\begin{aligned}&\left\| \rho _m(t)\right\| ^2+ 2\int _0^t[\left\| {\overline{\zeta }}_m\right\| ^2+\xi \left| A_1\rho _m\right| ^2_{L^2}+\langle B_1^0-B_1(u_m,\phi _m),A_1\rho _m\rangle ]ds\nonumber \\&\qquad +2\int _0^t[\xi \langle \zeta _m,A_1\rho _m\rangle +\xi \langle f^b-f(\phi _m) ,A_1\rho _m\rangle -\langle f^b-f(\phi _m) ,A_1\zeta _m\rangle ]ds\nonumber \\&\quad =2\sum _{j=1}^{m}\int _0^t (\varPhi _2(s)e_j-\sigma _1(s,u_m,\phi _m)e_j,\rho _m)_{H_2} d\beta ^j_s+2\sum _{j=m+1}^{+\infty }\int _0^t (\varPhi _2(s)e_j,\rho _m)_{H_2} d\beta ^j_s \nonumber \\&\qquad +\sum _{j=1}^{m}\int _0^t\left\| {\mathcal {P}}_m^2[ \varPhi _2(s)e_j-\sigma _2(s,u_m,\phi _m)e_j]\right\| ^2ds+\sum _{j=m+1}^{+\infty }\int _0^t\left\| {\mathcal {P}}_m^2 \varPhi _1(s)e_j\right\| ^2ds. \end{aligned}$$

(3.51)

Note that, owing to $\langle B_0(u_m,\theta _m),\theta _m \rangle =b_0(u_m,\theta _m,\theta _m)=0,$ we have

$$\begin{aligned} \langle B_0^0-B_0(u_m,u_m),\theta _m\rangle= & {} \langle B_0^0-B_0({\tilde{u}}_m,{\tilde{u}}_m),\theta _m\rangle +\langle B_0(\theta _m,{\widetilde{u}}_m),\theta _m\rangle \nonumber \\\le & {} \langle \beta _0^b-B_0({\tilde{u}}_m,{\tilde{u}}_m),\theta _m\rangle +\frac{1}{2}\left\| \theta _m\right\| ^2+c\left\| {\tilde{u}}_m\right\| ^2\left| \theta _m\right| ^2_{L^2}. \end{aligned}$$

(3.52)

Also we have

$$\begin{aligned} \int _0^t((\beta *\nabla \theta _m),\nabla \theta _m)ds&\ge 0, \end{aligned}$$

(3.53)

$$\begin{aligned} \langle B_1^0-B_1(u_m,\phi _m),A_1\rho _m\rangle&=\langle B_1^0-B_1({\tilde{u}}_m,{\tilde{\phi }}_m),A_1\rho _m\rangle \nonumber \\&\quad +\langle B_1(\theta _m,{\tilde{\phi }}_m),A_1\rho _m\rangle +\langle B_1(u_m,\rho _m),A_1\rho _m\rangle \nonumber \\&\le \langle B_1^0-B_1({\tilde{u}}_m,{\tilde{\phi }}_m),A_1\rho _m\rangle +\frac{1}{4}(\left\| \theta _m\right\| +\frac{\xi }{4}\left| A_1\rho _m\right| ^2_{L^2})\nonumber \\&\quad +c\left\| \phi _m\right\| ^2\left| A_1\phi _m\right| ^2_{L^2}\left| \theta _m\right| ^2_{L^2} +c\left| {\tilde{u}}_m\right| ^2_{L^2}\left\| {\tilde{u}}_m\right\| ^2\left\| \rho _m\right\| ^2, \end{aligned}$$

(3.54)

$$\begin{aligned} \langle R_0^0-R_0(A_1\phi _m,\phi _m),\theta _m\rangle&\le \langle R_0^0-R_0(A_1{\tilde{\phi }}_m,{\tilde{\phi }}_m),\theta _m\rangle +\frac{1}{4}(\left\| \theta _m\right\| +\frac{\xi }{4}\left| A_1\rho _m\right| ^2_{L^2})\nonumber \\&\quad + c\left| A_1\phi _m\right| ^2_{L^2}(\left| \theta _m\right| ^2_{L^2}+\left\| \rho _m\right\| ^2) + c\left\| {\tilde{\phi }}_m\right\| ^2\left| A_1{\tilde{\phi }}_m\right| ^2_{L^2}\left| \theta _m\right| ^2_{L^2}. \end{aligned}$$

(3.55)

Recall that from [18], there exists a monotone non-decreasing function $Q_1(x_1,x_2)$ such that

$$\begin{aligned} \langle f^b-f(\phi _m),A_1\zeta _m\rangle&\le \langle f^b-f({\tilde{\phi }}_m),A_1\zeta _m\rangle +\frac{1}{2}\left\| \zeta _m\right\| ^2 + Q_1(\left\| {\tilde{\phi }}_m\right\| ,\left\| \phi _m\right\| )(\left| A_1\phi _m\right| ^2_{L^2} +\left| A_1{\tilde{\phi }}_m\right| ^2_{L^2})\left\| \rho _m\right\| ^2, \nonumber \\ \xi \langle f^0-f(\phi _m),A_1\rho _m\rangle&\le \langle f^b-f({\tilde{\phi }}_m),A_1\rho _m\rangle +\frac{\xi }{8}\left| A_1\rho _m\right| ^2_{L^2}+Q_1(\left\| {\tilde{\phi }}_m\right\| ,\left\| \phi _m\right\| )\left\| \rho _m\right\| ^2, \nonumber \\ \xi \langle \zeta _m,A_1\rho _m\rangle&\le \frac{c\xi }{2}\left\| \zeta _m\right\| ^2+\frac{\xi }{4}\left| A_1\rho _m\right| ^2_{L^2}, \end{aligned}$$

(3.56)

$$\begin{aligned}&\sum _{j=1}^{m}\int _0^t\left| {\mathcal {P}}_m^1[ \varPhi _1(s)e_j-\sigma _1(s,u_m,\phi _m)e_j]\right| ^2_{L^2} ds\le \left\| {\mathcal {P}}_m^1(\varPhi _1(s)-\sigma _1(s,u_m,\phi _m))\right\| ^2_{{\mathcal {L}}^2(U;H_1)}\nonumber \\&\quad \le \left\| {\mathcal {P}}_m^1(\sigma _1(s,u,\phi )-\sigma _1(s,u_m,\phi _m))\right\| _{{\mathcal {L}}^2(U;H_1)} \nonumber \\&\qquad + 2 (({\mathcal {P}}_m^1(\varPhi _1(s)-\sigma _1(s,u_m,\phi _m)),{\mathcal {P}}_m^1(\varPhi _1(s)-\sigma _1(s,u,\phi ))))_{{\mathcal {L}}^2(U;H_1)}\nonumber \\&\qquad -\left\| {\mathcal {P}}_m^1(\sigma _1(s,u,\phi )-\varPhi _1(s))\right\| ^2_{{\mathcal {L}}^2(U;H_1)}\nonumber \\&\quad \le 2L^2\left| u-{\tilde{u}}_m\right| ^2_{L^2}+2L^2\left| (\theta _m,\rho _m)\right| ^2_{{\mathcal {H}}}\nonumber \\&\qquad + 2 (({\mathcal {P}}_m^1(\varPhi _1(s)-\sigma _1(s,u_m,\phi _m)),{\mathcal {P}}_m^1(\varPhi _1(s)-\sigma _1(s,u,\phi ))))_{{\mathcal {L}}^2(U;H_1)}\nonumber \\&\qquad -\left\| {\mathcal {P}}_m^1(\sigma _1(s,u,\phi )-\varPhi _1(s))\right\| ^2_{{\mathcal {L}}^2(U;H_1)}, \end{aligned}$$

(3.57)

$$\begin{aligned}&\sum _{j=1}^{m}\int _0^t\left\| {\mathcal {P}}_m^2[ \varPhi _2(s)e_j-\sigma _2(s,u_m,\phi _m)e_j]\right\| ^2 ds\le \left\| {\mathcal {P}}_m^2(\varPhi _2(s)-\sigma _2(s,u_m,\phi _m))\right\| ^2_{{\mathcal {L}}^2(U;H_2)}\nonumber \\&\quad \le 2L^2\left\| \phi -{\tilde{\phi }}_m\right\| ^2+2L^2\left| (\theta _m,\rho _m)\right| ^2_{{\mathcal {H}}}\nonumber \\&\qquad + 2 (({\mathcal {P}}_m^2(\varPhi _2(s)-\sigma _2(s,u_m,\phi _m)),{\mathcal {P}}_m^2(\varPhi _2(s)-\sigma _2(s,u,\phi ))))_{{\mathcal {L}}^2(U;H_1)}\nonumber \\&\qquad -\left\| {\mathcal {P}}_m^2(\sigma _2(s,u,\phi )-\varPhi _2(s))\right\| ^2_{{\mathcal {L}}^2(U;H_2)}, \end{aligned}$$

(3.58)

$$\begin{aligned} \Upsilon (s,z)&=\left| {\mathcal {P}}_m^1\gamma (s,u_m(s^-),\phi _m(s),z)-{\mathcal {P}}_m^1\varPsi (s,z)\right| ^2_{L^2}\nonumber \\&=\left| {\mathcal {P}}_m^1(\gamma (s,u(s^-),\phi (s),z)-\gamma (s,u_m(s^-))),{\mathcal {P}}_m^1(\phi _m(s),z))\right| ^2_{L^2}\nonumber \\&\quad +2({\mathcal {P}}_m^1(\varPsi (s,z)-\gamma (s,u_m(s^-),\phi _m(s),z)),{\mathcal {P}}_m^1(\varPsi (s,z)-\gamma (s,u(s^-),\phi (s),z)))\nonumber \\&\quad -\left| {\mathcal {P}}_m^1(\gamma (s,u(s^-),\phi (s),z)-\varPsi (s,z))\right| _{L^2}\nonumber \\&\le 2L^2\left| (u(s^-),\phi (s))-({\tilde{u}}_m(s^-),{\tilde{\phi }}_m(s))\right| _{\mathcal {H}}^2+2L^2\left| (\theta _m(s^-),\rho _m(s))\right| ^2_{{\mathcal {H}}}\nonumber \\&\quad -\left| {\mathcal {P}}_m^1(\gamma (s,u(s^-),\phi (s),z)-\varPsi (s,z))\right| _{L^2}+S_1(s,z), \end{aligned}$$

(3.59)

where

$$\begin{aligned} S_1(s,z)=2({\mathcal {P}}_m^1(\varPsi (s,z)-\gamma (s,u_m(s^-),\phi _m(s),z)),{\mathcal {P}}_m^1(\varPsi (s,z)-\gamma (s,u(s^-),\phi (s),z))). \end{aligned}$$

Let us set

$$\begin{aligned} Z(t)&=\left| \theta _m(t)\right| ^2_{L^2}+\left\| \rho _m(t)\right\| ^2,\nonumber \\ Y_1(t)&=c\left\| {\tilde{u}}_m\right\| ^2+c\left\| \phi _m\right\| ^2+\left| A_1\phi _m\right| ^2_{L^2}+c\left| {\tilde{u}}_m\right| ^2_{L^2}\left\| {\tilde{u}}_m\right\| ^2+c\left| A_1\phi _m\right| ^2_{L^2}\nonumber \\&\quad + c\left| A_1{\tilde{\phi }}_m\right| ^2_{L^2}\left\| {\tilde{\phi }}_m\right\| ^2+Q_1(\left\| {\tilde{\phi }}_m\right\| ,\left\| \phi _m\right\| )(1+\left| A_1\phi _m\right| ^2_{L^2}+\left| A_1{\tilde{\phi }}_m\right| ^2_{L^2})+4L^2,\nonumber \\ K_2(t)&=\left\| \theta _m\right\| ^2+(1-c\xi )\left\| \zeta _m\right\| ^2+c\xi \left| A_1\rho _m\right| ^2_{L^2}, \end{aligned}$$

(3.60)

where $\xi$ is small enough such that $1-c\xi >0.$ Also, let us set

$$\begin{aligned} \sigma (t)=\exp \left( -\int _0^tY_1(s)ds \right) . \end{aligned}$$

Adding (3.50) with (3.51), using (3.52)–(3.59), it follows from Itô’s formula that

$$\begin{aligned}&{\mathbb {E}}\sigma (t)Z(t)+{\mathbb {E}}\int _0^t\sigma (s)K_2(s)ds+{\mathbb {E}}\int _0^t\sigma (s)\left\| {\mathcal {P}}_m^1(\sigma _1(s,u,\phi )-\varPhi _1(s))\right\| ^2_{{\mathcal {L}}^2(U;H_1)}ds\nonumber \\&\qquad +{\mathbb {E}}\int _0^t\sigma (s)\left\| {\mathcal {P}}_m^2(\sigma _2(s,u,\phi )-\varPhi _2(s))\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds\nonumber \\&\qquad +{\mathbb {E}}\int _0^t\sigma (s)\left| {\mathcal {P}}_m^1(\gamma (s^-,u(s^-),\phi (s^-),z)-\varPsi (s,z))\right| ^2_{L^2}ds\nonumber \\&\quad \le {\mathbb {E}}\int _0^t\sigma (s)\langle -B_0^0+B_0({\tilde{u}}_m,{\tilde{u}}_m),\theta _m\rangle ds+ {\mathbb {E}}\int _0^t\sigma (s)\langle -B_1^0+B_1({\tilde{u}}_m,{\tilde{\phi }}_m),A_1\rho _m\rangle ds\nonumber \\&\qquad + 4L^2{\mathbb {E}}\int _0^t\sigma (s)\left| (u(s^-),\phi (s))-({\tilde{u}}_m(s^-),{\tilde{\phi }}_m(s))\right| _{\mathcal {H}}^2ds\nonumber \\&\qquad +\sum _{j=m+1}^{+\infty }{\mathbb {E}}\int _0^t\sigma (s)\left| {\mathcal {P}}_m^1 \varPhi _1(s)e_j\right| _{L^2}^2ds+\sum _{j=m+1}^{+\infty }{\mathbb {E}}\int _0^t\sigma (s)\left\| {\mathcal {P}}_m^2 \varPhi _2(s)e_j\right\| ^2ds\nonumber \\&\qquad + 2{\mathbb {E}}\int _0^t\sigma (s)((\varPhi _1(s)-\sigma _1(s,u_m,\phi _m),\varPhi _1(s)-\sigma _1(s,u,\phi )))_{{\mathcal {L}}^2(U;H_1)} ds\nonumber \\&\qquad +2{\mathbb {E}}\int _0^t\sigma (s)((\varPhi _2(s)-\sigma _2(s,u_m,\phi _m),\varPhi _2(s)-\sigma _2(s,u,\phi )))_{{\mathcal {L}}^2(U;H_2)} ds\nonumber \\&\qquad + {\mathbb {E}}\int _0^t\int _Z\sigma (s)S_1(s,z)\eta (dz,ds) + {\mathbb {E}}\int _0^t\sigma (s)\langle R_0^0-R_0(A_1{\tilde{\phi }}_m,{\tilde{\phi }}_m),\theta _m\rangle ds. \end{aligned}$$

(3.61)

Now, for each $n\ge 1,$ we consider the ${\mathcal {F}}_t$-stopping time $\tau _n$ defined by:

$$\begin{aligned} \tau _n=\min \left( T,\inf \left\{ t\in [0,T];\ \left\| (u,\phi )\right\| _{\mathcal {H}}^2+\int _0^t\left\| (u,\phi )\right\| _{\mathbb {V}}^2ds\ge n^2 \right\} \right) . \end{aligned}$$

We derive from (3.61) that

$$\begin{aligned}&{\mathbb {E}}\sigma (\tau _n)Z(\tau _n)+{\mathbb {E}}\int _0^{\tau _n}\sigma (s)K_2(s)ds+{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\left\| {\mathcal {P}}_m^1(\sigma _1(s,u,\phi )-\varPhi _1(s))\right\| ^2_{{\mathcal {L}}^2(U;H_1)}ds\nonumber \\&\qquad +{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\left\| {\mathcal {P}}_m^2(\sigma _2(s,u,\phi )-\varPhi _2(s))\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds\nonumber \\&\qquad +{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\left| {\mathcal {P}}_m^1(\gamma (s,u(s^-),\phi (s),z)-\varPsi (s,z))\right| ^2_{L^2}ds\nonumber \\&\quad \le 4L^2{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\left| (u(s^-),\phi (s))-({\tilde{u}}_m(s^-),{\tilde{\phi }}_m(s))\right| _{\mathcal {H}}^2ds\nonumber \\&\qquad +\sum _{j=m+1}^{+\infty }{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\left| {\mathcal {P}}_m^1 \varPhi _1(s)e_j\right| _{L^2}^2ds+\sum _{j=m+1}^{+\infty }{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\left\| {\mathcal {P}}_m^2 \varPhi _2(s)e_j\right\| ^2ds\nonumber \\&\qquad + 2{\mathbb {E}}\int _0^{\tau _n}\sigma (s)((\varPhi _1(s)-\sigma _1(s,u_m,\phi _m),\varPhi _1(s)-\sigma _1(s,u,\phi )))_{{\mathcal {L}}^2(U;H_1)} ds\nonumber \\&\qquad +2{\mathbb {E}}\int _0^{\tau _n}\sigma (s)((\varPhi _2(s)-\sigma _2(s,u_m,\phi _m),\varPhi _2(s)-\sigma _2(s,u,\phi )))_{{\mathcal {L}}^2(U;H_2)} ds\nonumber \\&\qquad +{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\langle -B_0^0+B_0({\tilde{u}}_m,{\tilde{u}}_m),\theta _m\rangle ds+ {\mathbb {E}}\int _0^{\tau _n}\sigma (s)\langle -B_1^0+B_1({\tilde{u}}_m,{\tilde{\phi }}_m),A_1\rho _m\rangle ds\nonumber \\&\qquad + {\mathbb {E}}\int _0^{\tau _n}\sigma (s)\langle R_0^0-R_0(A_1{\tilde{\phi }}_m,{\tilde{\phi }}_m),\theta _m\rangle ds+ {\mathbb {E}}\int _0^{\tau _n}\int _Z\sigma (s)S_1(s,z)\eta (dz,ds) . \end{aligned}$$

(3.62)

Now, we want to prove that the right side of (3.62) goes to 0 as m goes to $+\infty .$ We first note that, since $0<\sigma (t)\le 1$ and $({\tilde{u}}_m,{\tilde{\phi }}_m)\longrightarrow (u,\phi )$ in $L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;{\mathbb {V}}))$, we have

$$\begin{aligned}&\lim _{m\longrightarrow +\infty }\left( {\mathbb {E}}\int _0^{\tau _n}\sigma (s)\left| (u(s),\phi (s))-({\tilde{u}}_m(s),{\tilde{\phi }}_m(s))\right| _{\mathcal {H}}^2ds\right. \nonumber \\&\quad \left. +\sum _{j=m+1}^{+\infty }{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\left| {\mathcal {P}}_m^1 \varPhi _1(s)e_j\right| _{L^2}^2ds+\sum _{j=m+1}^{+\infty }{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\left\| {\mathcal {P}}_m^2 \varPhi _2(s)e_j\right\| ^2ds \right) =0. \end{aligned}$$

(3.63)

Following the same way as in [14, 36], we derive that

$$\begin{aligned}&\lim _{m\longrightarrow +\infty }{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\langle -B_0^0+B_0({\tilde{u}}_m,{\tilde{u}}_m),\theta _m\rangle ds=0,\\&\lim _{m\longrightarrow +\infty }{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\langle R_0^0-R_0(A_1{\tilde{\phi }}_m,{\tilde{\phi }}_m),\theta _m\rangle ds=0,\\&\lim _{m\longrightarrow +\infty }{\mathbb {E}}\int _0^{\tau _n}\sigma (s)\langle -B_1^0+B_1({\tilde{u}}_m,{\tilde{\phi }}_m),A_1\rho _m\rangle ds=0. \end{aligned}$$

Since ${\mathcal {P}}_m^i\circ {\mathcal {P}}_m^i={\mathcal {P}}_m^i$, and $\left\| {\mathcal {P}}_m^i\right\| \le 1$, $i=1,2,$ it follows that

$$\begin{aligned}&1_{[0,\tau _n]}\sigma (t){\mathcal {P}}_m^i(\Phi _i(s)-\sigma _i(s,u,\phi ))\in L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;{\mathcal {L}}^2(U;H_i)), \\&1_{[0,\tau _n]}\sigma (t){\mathcal {P}}_m^1(\varPsi (s,z)-\gamma (s,u,\phi ,z))\in {\mathcal {M}}^2_T(H_1). \end{aligned}$$

Therefore, as ${\mathcal {P}}_m^1\gamma (\cdot ,u_m(\cdot ),\phi _m(\cdot ))\rightharpoonup \Psi (.)$ in ${\mathcal {M}}^2_T(H_1)$ and ${\mathcal {P}}_m^i\sigma _i(\cdot ,u_m,\phi _m)\longrightarrow \varPhi _i(\cdot )$ in $L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;{\mathcal {L}}^2(U;H_i))),$ $i=1,2,$ we see that

$$\begin{aligned}&\lim _{m\longrightarrow +\infty } {\mathbb {E}}\int _0^{\tau _n}\int _Z\sigma (s)S_1(s,z)\eta (dz,ds) =0. \end{aligned}$$

(3.64)

$$\begin{aligned}&\lim _{m\longrightarrow +\infty }{\mathbb {E}}\int _0^{\tau _n}\sigma (s)(({\mathcal {P}}_m^i(\varPhi _i(s)-\sigma _i(s,u_m,\phi _m)),{\mathcal {P}}_m^i(\varPhi _i(s)-\sigma _i(s,u,\phi ))))_{{\mathcal {L}}^2(U;H_i)} ds=0,\quad i=1,2. \end{aligned}$$

(3.65)

This concludes that the right side of (3.62) goes to 0 as m goes to $+\infty .$

Now using the fact that $1_{[0,\tau _n]}\sigma (t)\le 1,$ we derive from (3.62) that

$$\begin{aligned}&\lim _{m\longrightarrow +\infty }{\mathbb {E}}\left| (\theta _m,\rho _m)\right| ^2_{\mathcal {H}}=\lim _{m\longrightarrow +\infty }{\mathbb {E}}\left| ({\tilde{u}}_m,{\tilde{\phi }})(\tau _n)-(\theta _m,\psi _m)(\tau _n)\right| ^2_{\mathcal {H}}=0\nonumber \\&\lim _{m\longrightarrow +\infty }{\mathbb {E}}\int _0^{\tau _n}K_2(s)ds=\lim _{m\longrightarrow +\infty }{\mathbb {E}}\int _0^{\tau _n}(\left\| \theta _m\right\| ^2+(1-c\xi )\left\| \zeta _m\right\| ^2+c\xi \left| A_1\rho _m\right| ^2_{L^2})ds\nonumber \\&=\lim _{m\longrightarrow +\infty }{\mathbb {E}}\int _0^{\tau _n}(\left\| {\widetilde{u}}_m-u_m\right\| ^2+(1-c\xi )\left\| {\tilde{\mu }}_m-\mu _m\right\| ^2+c\xi \left| A_1({\tilde{\phi }}_m-\phi _m)\right| ^2_{L^2})ds=0\nonumber \\&\lim _{m\longrightarrow +\infty }{\mathbb {E}}\int _0^{\tau _n}\left\| {\mathcal {P}}_m^1(\sigma _1(s,u,\phi )-\varPhi _1(s))\right\| ^2_{{\mathcal {L}}^2(U;H_1)}ds=0\nonumber \\&\lim _{m\longrightarrow +\infty }{\mathbb {E}}\int _0^{\tau _n}\left\| {\mathcal {P}}_m^2(\sigma _2(s,u,\phi )-\varPhi _2(s))\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds=0\nonumber \\&\lim _{m\longrightarrow +\infty }{\mathbb {E}}\int _0^{\tau _n}\left| {\mathcal {P}}_m^1(\gamma (s^-,u(s^-),\phi (s^-),z)-\varPsi (s,z))\right| _{L^2}^2ds=0. \end{aligned}$$

(3.66)

We note that from (3.66)$_{3,4,5}$ and the fact that the sequence $\{\tau _n;\ n \ge 1\}$ is increasing to T, we derive that

$$\begin{aligned}&\sigma _i(s,u,\phi )=\varPhi _i(s)\ \text { in }L^2(\Omega ,{\mathcal {F}},{\mathbb {P}};L^{2}(0,T;{\mathcal {L}}^2(U;H_i)),\ i=1,2, \\&\gamma (s,u(s^-),\phi (s),z)=\varPsi (s,z), \ \text { in }{\mathcal {M}}^2_T(H_1). \end{aligned}$$

The end of the proof of the Proposition 3.3 is very similar to [36, Proof of Claim 2]. $\square$

By Proposition 3.3, we infer from (3.43) that $(u,\phi )$ is a strong solution of problem (1.1) in the sense of Definition 2.1.

3.2 Uniqueness of Strong Solution

Assume that $(u_1,\phi _1)$ and $(u_2,\phi _2)$ are two strong solutions to (1.1). We set $(w,\psi ,\mu )=(u_1,\phi _1,\mu _1)-(u_2,\phi _2,\mu _2)$, ${\tilde{\sigma }}_i(\cdot \cdot )=\sigma _i(\cdot ,u_1(\cdot ),\phi _1(\cdot ))-\sigma _i(\cdot ,u_2(\cdot ),\phi _2(\cdot ))$, $i=1,2$ and ${\tilde{\gamma }}(\cdot ,\cdot )=\gamma (\cdot ,u_1(\cdot ),\phi _1(\cdot ),\cdot )-\gamma (\cdot ,u_2(\cdot ),\phi _2(\cdot ),\cdot )$. Then $(w,\psi )$ satisfies the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} d w+[A_0w+(\beta *A_0w)(t)+B_0(u_2,w)+B_0(w,u_1)]dt\\ =[R_0(A_1\phi _2,\psi )+R_0(A_1\psi ,\phi _1)]dt+{\tilde{\sigma }}_1(t)dW^1_t+\int _Z{\tilde{\gamma }}(t,z){\tilde{\pi }}(dt,dz)\ in\ V_1^{*},\\ d\psi +[A_1(\mu -\langle \mu \rangle )+B_1(u_2,\psi )+B_1(w,\phi _1)]dt=+{\tilde{\sigma }}_2(t)dW^2_t,\ in\ H^{-1}({\mathcal {M}}),\\ \mu =A_1\psi +f(\phi _1)-f(\phi _2),\\ (w,\psi )(0)=(0,0). \end{array}\right. } \end{aligned}$$

(3.67)

We apply infinite dimensional Itô’s formula (see [33]) to the process $\left| w\right| ^2_{L^2}$ and using the fact that $\left| x\right| ^2_{L^2}-\left| y\right| ^2_{L^2}+\left| x-y\right| ^2_{L^2}=2(x-y,x),$ $\forall x, y\in H_1$ to find that

$$\begin{aligned}&\left| w\right| ^2_{L^2}+2\int _0^t\left\| w(s)\right\| ^2ds=-2\int _0^t((\beta *\nabla w),\nabla w)ds-2\int ^t_0b_0(w,u_1,w)ds \nonumber \\&\quad +2\int ^t_0\langle R_0(A_1\phi _2,\psi )+R_0(A_1\psi ,\phi _1),w\rangle ds+\int ^t_0\int _Z\left| {\tilde{\gamma }}(s,z)\right| ^2_{L^2}\pi (ds,dz)\nonumber \\&\quad +\int _0^t\left\| {\tilde{\sigma }}_1(s)\right\| ^2_{{\mathcal {L}}^2(U;H_1)}ds+2\int ^t_0 ({\tilde{\sigma }}_1(s),w(s))dW^1_s\nonumber \\&\quad +2\int _0^t\int _Z({\tilde{\gamma }}(s,z),w(s^-)){\tilde{\pi }}(ds,dz). \end{aligned}$$

(3.68)

Also, applying the Itô formula to the process $\left\| \psi \right\| ^2$, we get

$$\begin{aligned} \left\| \psi \right\| ^2&=-2\int _0^t(A_1(\mu -\langle \mu \rangle ),A_1\psi )ds -2\int ^t_0b_1(w,\phi _1,A_1\psi )ds\nonumber \\&\quad -2\int ^t_0b_1(u_2,\psi ,A_1\psi ) ds+\int _0^t\left\| {\tilde{\sigma }}_2(s)\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds+2\int ^t_0 ({\tilde{\sigma }}_2(s),\psi (s))_{H_2}dW^2_s. \end{aligned}$$

(3.69)

Now we take the duality of (3.67)$_3$ with $A_1(\mu -\langle \mu \rangle ) -\xi A_1\psi$, where $\xi > 0$ is small enough and will be selected later. Adding the resulting equality to (3.68) and (3.69), we derive that

$$\begin{aligned} \left| (w,\psi )\right| ^2_{{\mathcal {H}}}&+2\int _0^t(\left\| w(s)\right\| ^2+\left\| \mu -\langle \mu \rangle \right\| ^2+\xi \left| A_1\psi \right| ^2_{L^2})ds=-2\int _0^t((\beta *\nabla w),\nabla w)ds\nonumber \\&-2\int ^t_0b_0(w,u_1,w)ds +2\int ^t_0\langle R_0(A_1\phi _2,\psi ),w\rangle ds-2\int ^t_0b_1(u_2,\psi ,A_1\psi ) ds\nonumber \\&+2\int _0^t[\xi (f(\phi _1)-f(\phi _2),A_1\psi )-(f(\phi _1)-f(\phi _2),A_1(\mu -\langle \mu \rangle ))]ds\nonumber \\&+2\int \xi (\mu -\langle \mu \rangle ,A_1\psi )ds+\int _0^t\left\| {\tilde{\sigma }}_1(s)\right\| ^2_{{\mathcal {L}}^2(U;H_1)}ds+2\int ^t_0 ({\tilde{\sigma }}_1(s),w(s))dW^1_s\nonumber \\&+\int _0^t\left\| {\tilde{\sigma }}_2(s)\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds+2\int ^t_0 ({\tilde{\sigma }}_2(s),\psi (s))_{H_2}dW^2_s\nonumber \\&+\int ^t_0\int _Z\left| {\tilde{\gamma }}(s,z)\right| ^2_{L^2}\pi (ds,dz)+2\int _0^t\int _Z({\tilde{\gamma }}(s,z),w(s^-)){\tilde{\pi }}(ds,dz). \end{aligned}$$

(3.70)

We note that

$$\begin{aligned} \left| b_0(w,u_1,w)\right|&\le \frac{1}{ 4}\left\| w\right\| ^2+c\left\| u_1\right\| ^2\left| w\right| ^2_{L^2}, \end{aligned}$$

(3.71)

$$\begin{aligned} \left| b_1(u_2,\psi ,A_1\psi )\right|&\le \frac{\xi }{8}\left| A_1\psi \right| ^2_{L^2}+c\left| u_2\right| ^2_{L^2}\left\| u_2\right\| ^2\left\| \psi \right\| ^2, \end{aligned}$$

(3.72)

$$\begin{aligned} \left| \langle R_0(A_1\phi _2,\psi ),w \rangle \right|&\le \frac{1}{4}(\left\| w\right\| ^2+\xi \left| A_1\psi \right| ^2_{L^2})+c(\left| w\right| ^2_{L^2}+\left| \nabla \psi \right| _{L^2}^2)\left\| \phi _2\right\| ^2\left| \phi _2\right| ^2_{H^2}, \end{aligned}$$

(3.73)

$$\begin{aligned} \xi \left| \langle f(\phi _1)-f(\phi _2), A_1\psi \rangle \right|&\le \frac{\xi }{16}\left| A_1\psi \right| ^2_{L^2}+Q_1(\left\| \phi _1\right\| ,\left\| \phi _2\right\| )\left\| \psi \right\| ^2, \end{aligned}$$

(3.74)

$$\begin{aligned} \left| \langle f(\phi _1)-f(\phi _2), \mu -\langle \mu \rangle \rangle \right|&\le \frac{\xi }{2}\left\| \mu -\langle \mu \rangle \right\| ^2\nonumber \\&\quad +Q_1(\left\| \phi _1\right\| ,\left\| \phi _2\right\| )(\left| A_1\phi _1\right| ^2_{L^2}+\left| A_1\phi _2\right| ^2_{L^2})\left\| \psi \right\| ^2, \end{aligned}$$

(3.75)

$$\begin{aligned} \xi \left| (\mu -\langle \mu \rangle ,A_1\psi )_{L^2}\right|&\le \frac{\xi }{16}\left| A_1\psi \right| ^2_{L^2}+c\xi \left| \nabla (\mu -\langle \mu \rangle )\right| ^2_{L^2}, \end{aligned}$$

(3.76)

$$\begin{aligned} \left\| {\tilde{\sigma }}_1(s)\right\| ^2_{{\mathcal {L}}^2(U;H_1)}+\left\| {\tilde{\sigma }}_2(s)\right\| ^2_{{\mathcal {L}}^2(U;H_2)}+\int _Z\left| {\tilde{\gamma }}(s,z)\right| ^2_{L^2}\lambda (dz)\le L\left| (w,\psi )\right| ^2_{{\mathcal {H}}}, \end{aligned}$$

(3.77)

where $Q_1$ is a suitable monotone non-decreasing function independent on time and the initial condition.

Now, let us set ${\mathcal {Y}}_2(t)=\left| w(t)\right| ^2_{L^2}+\left\| \psi \right\| ^2,$ and

$$\begin{aligned} K_1(t)&=c(\left\| u_1\right\| ^2+\left| u_2\right| ^2_{L^2}\left\| u_2\right\| ^2+\left\| \phi _2\right\| ^2\left| A_1\phi _2\right| ^2_{L^2}) \nonumber \\&\quad +Q_1(\left\| \phi _1\right\| ,\left\| \phi _2\right\| )(\left| A_1\phi _1\right| ^2_{L^2}+\left| A_1\phi _2\right| ^2_{L^2}), \end{aligned}$$

(3.78)

$$\begin{aligned} \sigma (t)=\exp \left( -\int _0^tK_1(s)ds \right) . \end{aligned}$$

So, applying Itô’s formula, to the real-valued process $\sigma (t){\mathcal {Y}}_2(t)$, using (3.70) and the inequalities (3.71)–(3.77), we derive that

$$\begin{aligned} {\mathbb {E}}\sigma (t){\mathcal {Y}}_2(t)&+{\mathbb {E}}\int _0^t\sigma (s)(\left\| w(s)\right\| ^2+(1-c\xi )\left\| (\mu -\langle \mu \rangle )\right\| ^2+\xi \left| A_1\psi \right| ^2_{L^2})ds\nonumber \\&+2{\mathbb {E}}\int _0^t\sigma (s)((\beta *\nabla w),\nabla w)ds\le L{\mathbb {E}}\int ^t_0\sigma (s){\mathcal {Y}}_2(s)ds,\qquad 0\le t\le T. \end{aligned}$$

Note that the expectation of the stochastic integrals in (3.70) varnishes. Therefore we obtain

$$\begin{aligned} {\mathbb {E}}\sigma (t){\mathcal {Y}}_2(t)\le L{\mathbb {E}}\int ^t_0\sigma (s){\mathcal {Y}}_2(s)ds,\qquad 0\le t\le T. \end{aligned}$$

It follows from the deterministic Gronwall lemma that ${\mathcal {Y}}_2(t)=0$ ${\mathbb {P}}$-a.s., for all $t\in [0,T].$ Hence $(u_1,\phi _1)=(u_2,\phi _2),$ ${\mathbb {P}}$-a.s., for all $t\in [0,T].$ Note that in (3.70), we choose $\xi >0$ and small enough such that $1-c\xi >0.$

4 Exponential Behavior

In this section, we show some aspects of the effects produced in the long-time behavior of the solution to a two dimensional Cahn–Hilliard–Oldroyd model with order one for the non-Newtonian two phase fluid flows under the presence of stochastic perturbations. More precisely, we discuss the moment exponential stability and almost sure exponential stability of strong solutions $(u,\phi )$ of stochastic 2D Cahn–Hilliard–Oldroyd model under some conditions.

We will consider the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} du(t)+ [\nu _1A_{0}u+(\beta *A_0u)(t)+B_{0}(u,u)-{\mathcal {K}}R_{0}(\varepsilon A_{1}\phi ,\phi )]dt\\ =g_1(t)dt+\sigma _1(t,u,\phi )dW^1_t +\int _Z\gamma (t,u(t^-),\phi (t),z){\tilde{\pi }}(dt,dz)dt,~ \text {in}~ V^*_{1}, \\ d\phi (t)+[\nu _2A_1\mu +B_{1}(u,\phi )]dt=g_2(t)dt+\sigma _2(t,u,\phi )dW^2_t,~ \text {in}~H^{-1}({\mathcal {M}}),\\ \mu =\varepsilon A_{1}\phi +\alpha f(\phi ) (u,\phi )(0)=(u_{0},\phi _{0}), \end{array}\right. } \end{aligned}$$

(4.1)

where $g=(g_1,g_2):[0,T]\rightarrow V_1^*\times H_2$ is Borel measurable function such that $g\in L^2(0,T;V_1^*\times H_2)$.

Remark 4.1

From the previous section, it is clear that for $g=(g_1,g_2)\in L^2(0,T;V_1^*\times H_2)$, there exists a unique (pathwise) global strong solution for the system (4.1) under the hypothesis (H1)–(H4).

Hereafter, as in [37], we assume that f satisfies the additional condition. For all $\phi _1,\phi _2\in D(A_1^{3/2})$,

$$\begin{aligned}&\langle \alpha A_1f(\phi _1)-\alpha A_1f(\phi _2), A_1\phi _1-A_1\phi _2 \rangle \ge -\alpha _0\left| A_1^{3/2}(\phi _1-\phi _2)\right| ^2_{L^2}, \end{aligned}$$

(4.2)

$$\begin{aligned}&\langle \alpha A_1f(\phi _1), A_1\phi _1 \rangle \ge -\alpha _0\left| A_1^{3/2}\phi _1\right| ^2_{L^2}, \end{aligned}$$

(4.3)

where $\alpha _0 > 0$ is a positive constants independent of $\phi _1$ and $\phi _2$.

Assuming that g is independent of t, we now consider the following stationary equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \left( \nu _1+ \frac{\gamma }{\delta } \right) A_{0}u^*+B_{0}(u^*,u^*)-{\mathcal {K}}R_{0}(\varepsilon A_{1}\phi ^*,\phi ^*) =g_1, \\ \nu _2\varepsilon A^2_{1}\phi ^*+\alpha A_1f(\phi ^*)+B_{1}(u^*,\phi ^*)=g_2. \end{array}\right. } \end{aligned}$$

(4.4)

Then we recall the following solvability result for the system (4.4) for $\nu =(\nu _1+ \frac{\gamma }{\delta } )>0$, where the proof is very similar to [37, Section 3.1].

Lemma 4.1

If $g=(g_1,g_2)\in V_1^*\times V_2^*$, then there exists a stationary solution $(u^*,\phi ^*)\in {\mathcal {U}}$ to system (4.4), Moreover, for $\varepsilon >0$ large enough such that $\alpha _2=\min ({\mathcal {K}}^{-1}\nu , \varepsilon ^2\nu _2-\varepsilon \alpha _0)$ is non negative, if $\alpha _2-2(\left\| g_1\right\| ^2_{V_1^*}+\left\| g_2\right\| _{V_2^*})>0$, then the stationary solution to (4.4) is unique.

Now, we give the definition of exponential stability.

Definition 4.1

We say that a strong solution $(u, \phi )(t)$ to (4.1) converges to $(u^*, \phi ^*)\in {\mathcal {H}}$ exponentially in the mean square if there exists $a>0$ and $M_0 = M_0((u, \phi )(0)) > 0$ such that

$$\begin{aligned} {\mathbb {E}}\left| (u,\phi )(t)-(u^*,\phi ^*)\right| ^2_{{\mathcal {H}}}\le M_0e^{-a t},\ \ t\ge 0. \end{aligned}$$

(4.5)

If $(u^*, \phi ^*)$ is a solution to (4.4), we say that $(u^*, \phi ^*)$ is exponentially stable in the mean square provided that every strong solution to (4.1) converges to $(u^*, \phi ^*)$ exponentially in the mean square with the same exponential order $a > 0.$

Theorem 4.1

Let $(u^*,\phi ^*)$ be the unique stationary solution of (4.4) and $\sigma _i(s,u^*,\phi ^*)=0$, $i=1,2$, $\gamma (s,u^*,\phi ^*,z)$, for all $s>0$ and $z\in Z$. Suppose that the assumption (H1)–(H5) are satisfied, then the strong solution $(u,\phi )(t)$ of system (4.1) converges to the stationary solution $(u^*,\phi ^*)$ of the system (4.4) is exponentially stable in the mean square provided that $\varepsilon$ is large enough such that $\alpha _2>0$ and the following inequality holds

$$\begin{aligned} \alpha _2 >2c_1\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}+\frac{\alpha _3L}{\lambda _0}, \end{aligned}$$

(4.6)

where $\alpha _2=\min ({\mathcal {K}}^{-1}\nu _1,\nu _2\varepsilon ^2-\alpha _0\varepsilon )$, $\alpha _3=\max ({\mathcal {K}}^{-1},\varepsilon )$ and $c_1>0$ is given below.

Proof

With the condition (4.6), one can chose a constant $a > 0$ such that

$$\begin{aligned} 0<a<\min \left\{ \delta , \lambda _0\left( \frac{\alpha _2}{2} -2c_1\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}-\frac{\alpha _3L}{2\lambda _0} \right) \right\} . \end{aligned}$$

(4.7)

We set $(w,\psi )(t)=(u,\phi )(t)-(u^*,\phi ^*).$

Applying the infinite dimensional Itô formula to the process ${\mathcal {K}}^{-1}e^{2a t}\left| w(t)\right| ^2_{L^2}$ we get

$$\begin{aligned}&{\mathcal {K}}^{-1}e^{2a t}\left| w(t)\right| ^2_{L^2}-{\mathcal {K}}^{-1}\left| w(0)\right| ^2_{L^2} \nonumber \\&\quad =2a\int _0^te^{2a s}{\mathcal {K}}^{-1}\left| w\right| ^2_{L^2}ds-2{\mathcal {K}}^{-1}\nu _1\int _0^te^{2a s}\langle A_0u,w \rangle ds\nonumber \\&\qquad -2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle (\beta *A_0u)(s),w \rangle ds -2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle B_0(u,u),w \rangle ds\nonumber \\&\qquad +2\int _0^te^{2a s}\langle R_0(\varepsilon A_1\phi ,\phi ),w \rangle ds +2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle g_1,w\rangle ds\nonumber \\&\qquad +{\mathcal {K}}^{-1}\int _0^te^{2a s}\left\| \sigma _1(s,u,\phi )\right\| ^2_{{\mathcal {L}}^2(U;H_1)}ds+2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle \sigma _1(s,u,\phi ),w\rangle dW^1_s\nonumber \\&\qquad +{\mathcal {K}}^{-1}\int _0^te^{2a s}\int _Z\left| \gamma (s,u(s),\phi (s),z)\right| ^2_{L^2}\pi (dz,ds)ds \nonumber \\&\qquad +2{\mathcal {K}}^{-1}\int _0^te^{2a s}\int _Z(\gamma (s,u(s^-),\phi (s),z),w){\tilde{\pi }}(dz,ds). \end{aligned}$$

(4.8)

Applying the Itô formula to the process $\varepsilon e^{2a t}\left| \nabla \psi (t)\right| ^2_{L^2},$ we derive that

$$\begin{aligned}&\varepsilon e^{2a t}\left| \nabla \psi (t)\right| ^2_{L^2} \nonumber \\&\quad =\varepsilon \left| \nabla \psi (0)\right| ^2_{L^2}+2a\varepsilon \int _0^te^{2a s}\left| \nabla \psi (s)\right| ^2_{L^2}ds -2\nu _2\varepsilon \int ^t_0e^{2a s}\langle A_1^2\phi ,\varepsilon A_1\psi \rangle ds\nonumber \\&\qquad -2\nu _2\alpha \int _0^te^{a s}\langle A_1f(\phi ),\varepsilon A_1\psi \rangle ds -2\int _0^te^{2a s}\langle B_1(u,\phi ),\varepsilon A_1\psi \rangle ds\nonumber \\&\qquad +2\varepsilon \int _0^te^{2a s}(g_2,A_1\psi )ds+\varepsilon \int _0^te^{2a s}\left\| \sigma _2(s,u,\phi )\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds\nonumber \\&\qquad +2\int _0^te^{2a s}( \sigma _2(s,u,\phi ),\psi )_{H_2} dW^2_s. \end{aligned}$$

(4.9)

Summing (4.8) and (4.9), after using (4.3), we derive that

$$\begin{aligned}&e^{2a t}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}\nonumber \\&\quad =\left| (w,\psi )(0)\right| ^2_{{\mathcal {H}}}+2a\int _0^te^{2a s}\left| (w,\psi )(s)\right| ^2_{{\mathcal {H}}}ds-2{\mathcal {K}}^{-1}\nu _1\int _0^te^{2a s}\langle A_0u,w \rangle ds\nonumber \\&\qquad -2\nu _2\varepsilon \int ^t_0e^{2a s}\langle A_1^2\phi ,\varepsilon A_1\psi \rangle ds-2\nu _2\alpha \int _0^te^{a s}\langle A_1f(\phi ),\varepsilon A_1\psi \rangle ds\nonumber \\&\qquad -2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle (\beta *A_0u)(s),w \rangle ds -2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle B_0(u,u),w \rangle ds\nonumber \\&\qquad +2\int _0^te^{2a s}\langle R_0(\varepsilon A_1\phi ,\phi ),w \rangle ds-2\int _0^te^{2a s}\langle B_1(u,\phi ),\varepsilon A_1\psi \rangle ds\nonumber \\&\qquad +2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle g_1,w\rangle ds+2\varepsilon \int _0^te^{2a s}(g_2,A_1\psi ) ds+\varepsilon \int _0^te^{2a s}\left\| \sigma _2(s,u,\phi )\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds\nonumber \\&\qquad +{\mathcal {K}}^{-1}\int _0^te^{2a s}\left\| \sigma _1(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_1)}ds+{\mathcal {K}}^{-1}\int _0^te^{2a s}\int _Z\left| \gamma (s,u(s),\phi (s),z)\right| ^2_{L^2}\pi (dz,ds) \nonumber \\&\qquad +2{\mathcal {K}}^{-1}\int _0^te^{2a s}\int _Z(\gamma (s,u(s^-),\phi (s),z),w){\tilde{\pi }}(dz,ds)\nonumber \\&\qquad +2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle \sigma _1(s,u,\phi ),w\rangle dW^1_s+2\int _0^te^{2a s}( \sigma _2(s,u,\phi ),\psi )_{H_2} dW^2_s. \end{aligned}$$

(4.10)

Using the definition of $\beta =\gamma e^{-\delta t}$, we note that

$$\begin{aligned} \int _0^te^{2a s}\langle \beta *A_0u^*,w(s) \rangle ds=\frac{\gamma }{\delta }\int _0^te^{2a s}(1-e^{-\delta s})\langle A_0u^*,w(s) \rangle ds. \end{aligned}$$

(4.11)

Using (4.11), we infer from (4.4) that $(u^*,\phi ^*)$ satisfies

$$\begin{aligned}&2{\mathcal {K}}^{-1}\nu _1\int _0^te^{2a s}\langle A_0u^*,w \rangle ds+2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle \beta *A_0u^*,w(s) \rangle ds\nonumber \\&\quad +\frac{2{\mathcal {K}}^{-1}\gamma }{\delta }\int _0^te^{(2a-\delta ) s}\langle A_0u^*,w(s) \rangle ds\nonumber \\&\quad +2\nu _2\varepsilon \int ^t_0e^{2a s}\langle A_1^2\phi ^*,\varepsilon A_1\psi \rangle ds +2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle B_0(u^*,u^*),w \rangle ds\nonumber \\&\quad -2\int _0^te^{2a s}\langle R_0(\varepsilon A_1\phi ^*,\phi ^*),w \rangle ds +2\int _0^te^{2a s}\langle B_1(u^*,\phi ^*),\varepsilon A_1\psi \rangle ds \nonumber \\&\quad +2\nu _1\alpha \int _0^te^{2a s}\langle A_1f(\phi ^*),\varepsilon A_1\psi \rangle ds =2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle g_1,w\rangle ds+2\varepsilon \int _0^te^{2a s} (g_2,A_1\psi )ds . \end{aligned}$$

(4.12)

Using (4.10) and (4.12), we derive that

$$\begin{aligned}&e^{2a t}{\mathbb {E}}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}} \nonumber \\&\quad ={\mathbb {E}}\left| (w,\psi )(0)\right| ^2_{{\mathcal {H}}}+2a\int _0^te^{2a s}{\mathbb {E}}\left| (w,\psi )(s)\right| ^2_{{\mathcal {H}}}ds\nonumber \\&\qquad -2{\mathcal {K}}^{-1}\nu _1\int _0^te^{2a s}{\mathbb {E}}\left\| w(s)\right\| ^2ds-2\nu _2\varepsilon ^2\int ^t_0e^{2a s} {\mathbb {E}}\left| A_1^{3/2}\psi (s) \right| ^2_{L^2}ds\nonumber \\&\qquad -2\nu _2\alpha {\mathbb {E}}\int _0^te^{a s}\langle A_1f(\phi )-A_1f(\phi ^*),\varepsilon A_1\psi \rangle ds-2{\mathcal {K}}^{-1}{\mathbb {E}}\int _0^te^{2a s}\langle (\beta *A_0w)(s),w \rangle ds\nonumber \\&\qquad +\frac{2{\mathcal {K}}^{-1}\gamma }{\delta }{\mathbb {E}}\int _0^te^{(2a-\delta ) s}\langle A_0u^*,w(s) \rangle ds-2{\mathcal {K}}^{-1}{\mathbb {E}}\int _0^te^{2a s}b_0(w,u^*,w) ds\nonumber \\&\qquad -2{\mathbb {E}}\int _0^te^{2a s}b_1(u^*,\psi ,\varepsilon A_1\psi ) ds+2{\mathbb {E}}\int _0^te^{2a s}b_1(w,\psi ,\varepsilon A_1\phi ^*) ds\nonumber \\&\qquad +\varepsilon {\mathbb {E}}\int _0^te^{2a s}\left\| \sigma _2(s,u,\phi )\right\| ^2_{{\mathcal {L}}^2(U;H_2)}ds +{\mathcal {K}}^{-1}{\mathbb {E}}\int _0^te^{2a s}\left\| \sigma _1(s,u,\phi )\right\| ^2_{{\mathcal {L}}^2(U;H_1)}ds\nonumber \\&\qquad +{\mathcal {K}}^{-1}{\mathbb {E}}\int _0^te^{2a s}\int _Z\left| \gamma (s,u(s),\phi (s),z)\right| ^2_{L^2}\lambda (dz)ds. \end{aligned}$$

(4.13)

Using (2.30), we have

$$\begin{aligned} {\mathbb {E}}\int _0^te^{2a s}\langle (\beta *A_0w)(s),w \rangle ds={\mathbb {E}}\int _0^te^{2a s} ((\beta *\nabla w)(s),\nabla w) ds\ge 0. \end{aligned}$$

(4.14)

Using Cauchy-Schwarz, Hölder’s and Young’s inequalities, we get

$$\begin{aligned} {\mathbb {E}}\int _0^te^{(2a-\delta ) s}\langle A_0u^*,w(s) \rangle ds&\le {\mathbb {E}}\int _0^te^{(2a-\delta ) s}\left\| u^*\right\| \left\| w(s)\right\| ds\nonumber \\&\le \left( \int _0^te^{2(a-\delta ) s}\left\| u^*\right\| ^2 ds\right) ^{1/2}\left( {\mathbb {E}}\int _0^te^{2a s}\left\| w(s)\right\| ^2 ds\right) ^{1/2}\nonumber \\&\le \frac{\alpha _2\delta }{2{\mathcal {K}}^{-1}\gamma }{\mathbb {E}}\int _0^te^{2a s}\left\| w(s)\right\| ^2 ds+\frac{{\mathcal {K}}^{-1}\gamma }{4\alpha _2\delta (\delta -a)}\left\| u^*\right\| ^2, \end{aligned}$$

(4.15)

for $0<a<\delta$.

Note that

$$\begin{aligned} {\mathcal {K}}^{-1}\left| b_0(w,u^*,w)\right|&\le c_1\left\| u^*\right\| \left\| w\right\| ^2 \end{aligned}$$

(4.16)

$$\begin{aligned} \left| b_1(w,\psi ,\varepsilon A_1\phi ^*) \right|&\le \varepsilon c\left| A_1\psi \right| _{L^2}\left| A_1\phi ^*\right| _{L^2}\left\| w\right\| \nonumber \\&\le c_1\left| A^{3/2}_1\phi ^*\right| _{L^2}(\left\| w\right\| ^2+\left| A^{3/2}_1\psi \right| _{L^2}^2) \end{aligned}$$

(4.17)

$$\begin{aligned} \left| b_1(u^*,\psi ,\varepsilon A_1\psi )\right|&\le c_1\left\| u^*\right\| \left| A_1^{3/2}\psi \right| _{L^2}^2 \end{aligned}$$

(4.18)

$$\begin{aligned} \nu _2\langle \alpha A_1f(\phi )-\alpha A_1f(\phi ^*),\varepsilon A_1\psi \rangle&\ge -\alpha _0\nu _2\varepsilon \left| A_1^{3/2}\psi \right| ^2_{L^2}. \end{aligned}$$

(4.19)

Using (2.39), (4.16)–(4.19) and (2.23) in (4.13) we get

$$\begin{aligned}&e^{2a t}{\mathbb {E}}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}+2\left( \frac{\alpha _2}{2}-\frac{(2a+\alpha _3L)}{2\lambda _0}-c_1\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}} \right) \int _0^te^{2a s}{\mathbb {E}}\left\| (w,\psi )(s)\right\| ^2_{{\mathcal {U}}}ds\nonumber \\&\quad \le {\mathbb {E}}\left| (w,\psi )(0)\right| ^2_{{\mathcal {H}}}+\frac{{\mathcal {K}}^{-2}\gamma ^2}{2\alpha _2\delta ^2(\delta -a)}\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}^2. \end{aligned}$$

(4.20)

Since a satisfies (4.7), we finally have

$$\begin{aligned} {\mathbb {E}}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}\le e^{-2a t}\left[ {\mathbb {E}}\left| (w,\psi )(0)\right| ^2_{{\mathcal {H}}}+\frac{{\mathcal {K}}^{-2}\gamma ^2}{2\alpha _2\delta ^2(\delta -a)}\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}^2\right] , \end{aligned}$$

(4.21)

and hence $(u,\phi )(t)$ converges to $(u^*,\phi ^*)$ exponentially in the mean square. $\square$

Theorem 4.2

Suppose that all conditions given in Theorem 4.1 are satisfied, then the strong solution $(u, \phi )(t)$ of (4.1) converges to the stationary solution $(u^*, \phi ^*)$ of (4.4) almost surely exponentially.

Proof

Let $n=1,2,\ldots .,$ and $h>0$. By the Itô formula, for any $t \ge N$ we have

$$\begin{aligned}&\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}} \nonumber \\&\quad =\left| (w,\psi )(nh)\right| ^2_{{\mathcal {H}}}-2{\mathcal {K}}^{-1}\nu _1\int _{nh}^t\left\| w(s)\right\| ^2ds-2\nu _2\varepsilon ^2\int ^t_{nh} \left| A_1^{3/2}\psi (s) \right| ^2_{L^2}ds\nonumber \\&\qquad -2\nu _2\alpha \int _{nh}^t\langle A_1f(\phi )-A_1f(\phi ^*),\varepsilon A_1\psi \rangle ds-2{\mathcal {K}}^{-1}\int _{nh}^t\langle (\beta *A_0w)(s),w \rangle ds\nonumber \\&\qquad +\frac{2{\mathcal {K}}^{-1}\gamma }{\delta }\int _{nh}^te^{-\delta s}\langle A_0u^*,w(s) \rangle ds-2{\mathcal {K}}^{-1}\int _{nh}^tb_0(w,u^*,w) ds\nonumber \\&\qquad -2\int _{nh}^tb_1(u^*,\psi ,\varepsilon A_1\psi ) ds+2\int _{nh}^tb_1(w,\psi ,\varepsilon A_1\phi ^*) ds\nonumber \\&\qquad +\varepsilon \int _{nh}^t\left\| \sigma _2(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_2)}ds +{\mathcal {K}}^{-1}\int _{nh}^t\left\| \sigma _1(s,u,\phi )\right\| ^2_{{\mathcal {L}}^2(U;H_1)}ds\nonumber \\&\qquad +{\mathcal {K}}^{-1}\int _{nh}^t\int _Z\left| \gamma (s,u(s),\phi (s),z)\right| ^2_{L^2}\lambda (dz)ds\nonumber \\&\qquad +2{\mathcal {K}}^{-1}\int _{nh}^t\langle \sigma _1(s,u,\phi ),w\rangle dW^1_s+2\varepsilon \int _{nh}^t( \sigma _2(s,u,\phi ),\psi )_{H_2} dW^2_s. \end{aligned}$$

(4.22)

Taking supremum from nh to $(n+1)h$ and then taking expectation in (4.22) after using (4.14)–(4.19) with $a=1$, we find

$$\begin{aligned}&{\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}+\alpha _2{\mathbb {E}}\int _{nh}^{(n+1)h}\left\| (w,\psi )(s)\right\| ^2_{\mathcal {U}}ds\le {\mathbb {E}}\left| (w,\psi )(nh)\right| ^2_{{\mathcal {H}}}\nonumber \\&\quad +\frac{{\mathcal {K}}^{-2}\gamma ^2}{2\alpha _2\delta ^3}\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}^2e^{-2\delta nh} +2c_1\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}{\mathbb {E}}\int _{nh}^{(n+1)h}\left\| (w,\psi )(s)\right\| ^2_{\mathcal {U}}ds\nonumber \\&\quad +\varepsilon {\mathbb {E}}\int _{nh}^{(n+1)h}\left\| \sigma _2(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_2)}ds +{\mathcal {K}}^{-1}{\mathbb {E}}\int _{nh}^{(n+1)h}\left\| \sigma _1(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_1)}ds\nonumber \\&\quad +{\mathcal {K}}^{-1}{\mathbb {E}}\int _{nh}^{(n+1)h}\int _Z\left| \gamma (s,u(s),\phi (s),z)\right| ^2_{L^2}\lambda (dz)ds\nonumber \\&\quad +2{\mathcal {K}}^{-1}{\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| \int _{nh}^t\int _Z(\gamma (s,u(s^-),\phi (s),z),w){\tilde{\pi }}(dz,ds)\right| \nonumber \\&\quad +2{\mathcal {K}}^{-1}{\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| \int _{nh}^t\langle \sigma _1(s,u,\phi ),w\rangle dW^1_s\right| \nonumber \\&\quad +2\varepsilon {\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| \int _{nh}^t( \sigma _2(s,u,\phi ),\psi )_{H_2} dW^2_s\right| . \end{aligned}$$

(4.23)

By, Davis’, Hölder’s, and Young’s inequalities we derive

$$\begin{aligned}&{\mathcal {K}}^{-1}{\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| \int _{nh}^t\langle \sigma _1(s,u,\phi ),w\rangle dW^1_s\right| +\varepsilon {\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| \int _{nh}^t( \sigma _2(s,u,\phi ),\psi )_{H_2} dW^2_s\right| \nonumber \\&\quad \le c{\mathcal {K}}^{-1}{\mathbb {E}}\left( \int _{nh}^{(n+1)h} \left\| \sigma _1(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_1)}\left| w\right| _{L^2}^2ds\right) ^{1/2}\nonumber \\&\qquad +c\varepsilon {\mathbb {E}}\left( \int _{nh}^{(n+1)h} \left\| \sigma _2(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_2)}\left\| \phi \right\| ^2ds\right) ^{1/2}\nonumber \\&\quad \le \frac{1}{8}{\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}+c_2\sum _{i=0}^2{\mathbb {E}}\int _{nh}^{(n+1)h} \left\| \sigma _i(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_i)}ds \end{aligned}$$

(4.24)

An application of the Burkholder–Davis–Gundy inequality (see [33, Theorem 48]), Hölder’s, and Young’s inequalities yield

$$\begin{aligned}&{\mathcal {K}}^{-1}{\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| \int _{nh}^t\int _Z(\gamma (s,u(s^-),\phi (s),z),w){\tilde{\pi }}(dz,ds)ds\right| \nonumber \\&\quad \le c2{\mathcal {K}}^{-1}{\mathbb {E}}\left( \int _{nh}^{(n+1)h}\int _Z\left| \gamma (s,u(s),\phi (s),z)\right| ^2_{L^2}\left| w\right| ^2_{L^2}\pi (dz,ds)\right) ^{1/2}\nonumber \\&\quad \le c2{\mathcal {K}}^{-1}{\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| w\right| _{L^2}\left( \int _{nh}^{(n+1)h}\int _Z\left| \gamma (s,u(s),\phi (s),z)\right| ^2_{L^2}\pi (dz,ds)\right) ^{1/2}\nonumber \\&\quad \le \frac{1}{8}{\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}+c_2{\mathbb {E}}\int _{nh}^{(n+1)h}\int _Z\left| \gamma (s,u(s),\phi (s),z)\right| ^2_{L^2}\lambda (dz)ds. \end{aligned}$$

(4.25)

Combining (4.24) and (4.25), substituting it in (4.23), and then using (2.39), we get

$$\begin{aligned}&{\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}+\alpha _4{\mathbb {E}}\int _{nh}^{(n+1)h}\left\| (w,\psi )(s)\right\| ^2_{\mathcal {U}}ds\nonumber \\&\quad \le 2{\mathbb {E}}\left| (w,\psi )(nh)\right| ^2_{{\mathcal {H}}}+\frac{{\mathcal {K}}^{-2}\gamma ^2}{\alpha _2\delta ^3}\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}^2e^{-2\delta nh}, \end{aligned}$$

(4.26)

where

$$\begin{aligned} \alpha _3=2\lambda _0\left( \alpha _2-2c_1\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}+\frac{c_2L}{\lambda _0}\right) >0. \end{aligned}$$

Using (4.21) in (4.26), we arrive at

$$\begin{aligned} {\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}\le \left( 2{\mathbb {E}}\left| (w,\psi )(0)\right| ^2_{{\mathcal {H}}}+\frac{{\mathcal {K}}^{-2}\gamma ^2(2\delta -a)}{\alpha _2\delta ^3(\delta -a)}\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}^2\right) e^{-2a nh}, \end{aligned}$$

(4.27)

For $\theta \in (0,a)$, we set

$$\begin{aligned} \Omega ^a_{n,h}=\left\{ \omega \in \Omega : \sup _{nh\le t\le (n+1)h}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}>e^{-(a-\theta ) nh} \right\} . \end{aligned}$$

By Chebychev’s inequality, we also have

$$\begin{aligned} {\mathbb {P}}(\Omega ^a_{n,h})&\le e^{2(a-\theta ) nh}{\mathbb {E}}\sup _{nh\le t\le (n+1)h}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}\nonumber \\&\le \left( 2{\mathbb {E}}\left| (w,\psi )(0)\right| ^2_{{\mathcal {H}}}+\frac{{\mathcal {K}}^{-2}\gamma ^2(2\delta -a)}{\alpha _2\delta ^3(\delta -a)}\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}^2\right) e^{-2\theta nh}, \end{aligned}$$

(4.28)

which implies that

$$\begin{aligned} \sum _{n=1}^\infty {\mathbb {P}}(\Omega ^a_{n,h})\le \left( 2{\mathbb {E}}\left| (w,\psi )(0)\right| ^2_{{\mathcal {H}}}+\frac{{\mathcal {K}}^{-2}\gamma ^2(2\delta -a)}{\alpha _2\delta ^3(\delta -a)}\left\| (u^*,\phi ^*)\right\| _{{\mathcal {U}}}^2\right) \frac{1}{e^{2\theta h}-1}<+\infty . \end{aligned}$$

Therefore, by the Borel-Cantelli lemma, there is a finite integer $n_0(\omega )$ such that

$$\begin{aligned} \sup _{nh\le t\le (n+1)h}\left| (w,\psi )(t)\right| ^2_{{\mathcal {H}}}\le e^{-(a-\theta ) nh},\quad {\mathbb {P}}-a.s., \end{aligned}$$

(4.29)

for all $n\ge n_0$, and the Theorem 4.2 is then proved. $\square$

For the next theorem we assume that $g_1$ and $g_2$ depend on $u(\cdot )$, $\phi (\cdot )$ and satisfy the following Lipschitz condition: For all $(u_1,\phi _1), (u_2,\phi _2)\in {\mathcal {U}}$,

$$\begin{aligned} \begin{aligned} \left\| g_1(u_1,\phi _2)-g_1(u_1,\phi _2)\right\| _{V_1^*}\le L_1\left\| (u_1,\phi _1)-(u_2,\phi _2)\right\| _{\mathcal {U}},\\ \left\| g_2(u_1,\phi _1)-g_2(u_2,\phi _2)\right\| _{H_2}\le L_2\left| (u_1,\phi _1)-(u_2,\phi _2)\right| _{\mathcal {H}}. \end{aligned} \end{aligned}$$

(4.30)

Theorem 4.3

If $g_i(0,0)=0$, $\sigma _i(t,0,0)=0$, $i=1,2$ and $\gamma (t,0,0,z)=0$, for all $t> 0$ and $z\in Z$, then any strong solution $(u,\phi )(t)$ to (4.1) converges to zero almost surely exponentially if

$$\begin{aligned} \alpha _2>{\mathcal {K}}^{-1}L_1-\frac{(L_2+\alpha _3L)}{2\lambda _0}. \end{aligned}$$

(4.31)

Proof

Owing to (4.31), one can chose a constant $a > 0$ such that

$$\begin{aligned} 0<a<\lambda _0\left( \alpha _2-{\mathcal {K}}^{-1}L_1-\frac{(L_2+\alpha _3L)}{2\lambda _0}\right) . \end{aligned}$$

(4.32)

Applying the infinite dimensional Itô formula to the process ${\mathcal {K}}^{-1}e^{2a t}\left| u(t)\right| ^2_{L^2}$ and $\varepsilon e^{2a t}\left| \nabla \psi (t)\right| ^2_{L^2}$ respectively, summing the results and using (2.7)–(2.8), we get

$$\begin{aligned}&e^{2a t}\left| (u,\phi )(t)\right| ^2_{{\mathcal {H}}} \nonumber \\&\quad =\left| (u,\phi )(0)\right| ^2_{{\mathcal {H}}}+2a\int _0^te^{2a s}\left| (u,\phi )(s)\right| ^2_{{\mathcal {H}}}ds-2{\mathcal {K}}^{-1}\nu _1\int _0^te^{2a s}\langle A_0u,u \rangle ds\nonumber \\&\qquad -2\nu _2\varepsilon \int ^t_0e^{2a s}\langle A_1^2\phi ,\varepsilon A_1\phi \rangle ds-2\nu _2\alpha \int _0^te^{a s}\langle A_1f(\phi ),\varepsilon A_1\phi \rangle ds\nonumber \\&\qquad -2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle (\beta *A_0u)(s),u \rangle ds+2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle g_1(u,\phi ),u\rangle ds\nonumber \\&\qquad +2\varepsilon \int _0^te^{2a s}(\nabla g_2(u,\phi ),\nabla \phi ) ds+\varepsilon \int _0^te^{2a s}\left\| \sigma _2(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_2)}ds\nonumber \\&\qquad +{\mathcal {K}}^{-1}\int _0^te^{2a s}\left\| \sigma _1(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_1)}ds\nonumber \\&\qquad +{\mathcal {K}}^{-1}\int _0^te^{2a s}\int _Z\left| \gamma (s,u(s),\phi (s),z)\right| ^2_{L^2}\pi (dz,ds) \nonumber \\&\qquad +2{\mathcal {K}}^{-1}\int _0^te^{2a s}\int _Z(\gamma (s,u(s^-),\phi (s),z),u){\tilde{\pi }}(dz,ds)\nonumber \\&\qquad +2{\mathcal {K}}^{-1}\int _0^te^{2a s}\langle \sigma _1(s,u,\phi ),u\rangle dW^1_s+2\int _0^te^{2a s}( \sigma _2(s,u,\phi ),\psi )_{H_2} dW^2_s. \end{aligned}$$

(4.33)

Using (2.30), (4.3), (2.39) and (4.30), we infer from (4.33) that

$$\begin{aligned}&e^{2a t}{\mathbb {E}}\left| (u,\phi )(t)\right| ^2_{{\mathcal {H}}}+2\alpha _2{\mathbb {E}}\int _0^te^{2a s}\left\| (u,\phi )(s)\right\| ^2_{{\mathcal {U}}}ds\le {\mathbb {E}}\left| (u,\phi )(0)\right| ^2_{{\mathcal {H}}}+2a{\mathbb {E}}\int _0^te^{2a s}\left| (u,\phi )(s)\right| ^2_{{\mathcal {H}}}ds\\&\qquad +2{\mathcal {K}}^{-1}{\mathbb {E}}\int _0^te^{2a s}\langle g_1(u,\phi ),u\rangle ds+2\varepsilon {\mathbb {E}}\int _0^te^{2a s}(\nabla g_2(u,\phi ),\nabla \phi ) ds+\varepsilon {\mathbb {E}}\int _0^te^{2a s}\left\| \sigma _2(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_2)}ds\\&\qquad +{\mathcal {K}}^{-1}{\mathbb {E}}\int _0^te^{2a s}\left\| \sigma _1(s,u,\phi )\right\| ^2_{{\mathcal {L}}(U;H_1)}ds+{\mathcal {K}}^{-1}{\mathbb {E}}\int _0^te^{2a s}\int _Z\left| \gamma (s,u(s^-),\phi (s),z)\right| ^2_{L^2}\lambda (dz)ds \\&\quad \le {\mathbb {E}}\left| (u,\phi )(0)\right| ^2_{{\mathcal {H}}}+2{\mathcal {K}}^{-1}L_1{\mathbb {E}}\int _0^te^{2a s}\left\| (u,\phi )(s)\right\| ^2_{{\mathcal {U}}}ds+(2a+L_2+\alpha _3L){\mathbb {E}}\int _0^te^{2a s}\left| (u,\phi )(s)\right| ^2_{{\mathcal {H}}}ds. \end{aligned}$$

This implies that

$$\begin{aligned} e^{2a t}{\mathbb {E}}\left| (u,\phi )(t)\right| ^2_{{\mathcal {H}}}+2\left( \alpha _2-{\mathcal {K}}^{-1}L_1-\frac{(2a+L_2+\alpha _3L)}{2\lambda _0} \right) {\mathbb {E}}\int _0^te^{2a s}\left\| (u,\phi )(s)\right\| ^2_{{\mathcal {U}}}ds\le {\mathbb {E}}\left| (u,\phi )(0)\right| ^2_{{\mathcal {H}}}. \end{aligned}$$

under the condition (4.31), it is immediate that

$$\begin{aligned} {\mathbb {E}}\left| (u,\phi )(t)\right| ^2_{{\mathcal {H}}}\le {\mathbb {E}}\left| (u,\phi )(0)\right| ^2_{{\mathcal {H}}}e^{-2a t}. \end{aligned}$$

This implies that the strong solution of (4.1) converges to zero exponentially in the mean square. We can then finish the proof using the same method as in the proof of Theorem 4.2. $\square$